Polynomial decay of correlations for flows,
including Lorentz gas examples

Péter Bálint MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Egry József u. 1, H-1111 Budapest, Hungary and Department of Stochastics, Budapest University of Technology and Economics Egry József u. 1, H-1111, Budapest , Hungary pet@math.bme.hu    Oliver Butterley ICTP - Strada Costiera, 11 I-34151 Trieste, Italy. oliver.butterley@ictp.it    Ian Melbourne Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. i.melbourne@warwick.ac.uk
(6 October 2017; revised 10 November 2018)
Abstract

We prove sharp results on polynomial decay of correlations for nonuniformly hyperbolic flows. Applications include intermittent solenoidal flows and various Lorentz gas models including the infinite horizon Lorentz gas.

1 Introduction

Let (Λ,μΛ)Λsubscript𝜇Λ(\Lambda,\mu_{\Lambda}) be a probability space. Given a measure-preserving flow Tt:ΛΛ:subscript𝑇𝑡ΛΛT_{t}:\Lambda\to\Lambda and observables v,wL2(Λ)𝑣𝑤superscript𝐿2Λv,w\in L^{2}(\Lambda), we define the correlation function ρv,w(t)=ΛvwTt𝑑μΛΛv𝑑μΛΛw𝑑μΛsubscript𝜌𝑣𝑤𝑡subscriptΛ𝑣𝑤subscript𝑇𝑡differential-dsubscript𝜇ΛsubscriptΛ𝑣differential-dsubscript𝜇ΛsubscriptΛ𝑤differential-dsubscript𝜇Λ\rho_{v,w}(t)=\int_{\Lambda}v\;w\circ T_{t}\,d\mu_{\Lambda}-\int_{\Lambda}v\,d\mu_{\Lambda}\int_{\Lambda}w\,d\mu_{\Lambda}. The flow is mixing if limtρv,w(t)=0subscript𝑡subscript𝜌𝑣𝑤𝑡0\lim_{t\to\infty}\rho_{v,w}(t)=0 for all v,wL2(Λ)𝑣𝑤superscript𝐿2Λv,w\in L^{2}(\Lambda).

Of interest is the rate of decay of correlations, or rate of mixing, namely the rate at which ρv,wsubscript𝜌𝑣𝑤\rho_{v,w} converges to zero. Dolgopyat [17] showed that geodesic flows on compact surfaces of negative curvature with volume measure μΛsubscript𝜇Λ\mu_{\Lambda} are exponentially mixing for Hölder observables v,w𝑣𝑤v,w. Liverani [22] extended this result to arbitrary dimensional geodesic flows in negative curvature and more generally to contact Anosov flows. However, exponential mixing remains poorly understood in general.

Dolgopyat [18] considered the weaker notion of rapid mixing (superpolynomial decay of correlations) where ρv,w(t)=O(tq)subscript𝜌𝑣𝑤𝑡𝑂superscript𝑡𝑞\rho_{v,w}(t)=O(t^{-q}) for sufficiently regular observables for any fixed q1𝑞1q\geq 1, and showed that rapid mixing is ‘prevalent’ for Axiom A flows: it suffices that the flow contains two periodic solutions with periods whose ratio is Diophantine. Field et al. [19] introduced the notion of good asymptotics and used this to prove that amongst Crsuperscript𝐶𝑟C^{r} Axiom A flows, r2𝑟2r\geq 2, an open and dense set of flows is rapid mixing.

In [24], results on rapid mixing were obtained for nonuniformly hyperbolic semiflows, combining the rapid mixing method of Dolgopyat [18] with advances by Young [30, 31] in the discrete time setting. First results on polynomial mixing for nonuniformly hyperbolic semiflows (ρv,w(t)=O(tq)subscript𝜌𝑣𝑤𝑡𝑂superscript𝑡𝑞\rho_{v,w}(t)=O(t^{-q}) for some fixed q>0𝑞0q>0) were obtained in [25]. Under certain assumptions the results in [24, 25] were established also for nonuniformly hyperbolic flows. However, for polynomially mixing flows, the assumptions in [25] are overly restrictive and exclude many examples including infinite horizon Lorentz gases.

In this paper, we develop the tools required to cover systematically large classes of nonuniformly hyperbolic flows. The recent review article [26] describes the current state of the art for rapid and polynomial decay of correlations for nonuniformly hyperbolic semiflows and flows and gives a complete self-contained proof in the case of semiflows. Here we provide the arguments required to deal with flows. Our results cover all of the examples in [26].

By [24], rapid mixing holds (at least typically) for nonuniformly hyperbolic flows that are modelled as suspensions over Young towers with exponential tails [30]. See also Remark 8.5. Here we give a different proof that has a number of advantages as discussed in the introduction to [26]. Flows are modelled as suspensions over a uniformly hyperbolic map with an unbounded roof function (rather than as suspensions over a nonuniformly hyperbolic map with a bounded roof function). It then suffices to consider twisted transfer operators with one complex parameter rather than two as in [24], reducing from four to three the number of periodic orbits that need to be considered in Proposition 6.6. Also, the proof of rapid mixing only uses superpolynomial tails for the roof function, whereas [24] requires exponential tails.

Examples covered by our results on rapid mixing include finite Lorentz gases (including those with cusps, corner points, and external forcing), Lorenz attractors, and Hénon-like attractors. We refer to [26] for references and further details.

Examples discussed in [25, 26] for which polynomial mixing holds include nonuniformly hyperbolic flows that are modelled as suspensions over Young towers with polynomial tails [31]. This includes intermittent solenoidal flows, see also Remark 8.6.

The key example of continuous time planar periodic infinite horizon Lorentz gases is considered at length in Section 9. In the finite horizon case, exponential decay of correlations for the flow was proved in [4]. In the infinite horizon case it has been conjectured [20, 23] that the decay rate for the flow is O(t1)𝑂superscript𝑡1O(t^{-1}). (An elementary argument in [5] shows that this rate is optimal; the argument is reproduced in the current context in Proposition 9.14.) We obtain the conjectured decay rate O(t1)𝑂superscript𝑡1O(t^{-1}) for planar infinite horizon Lorentz flows in Theorem 9.1.

Remark 1.1

(a) In [25], the decay rate O(t1)𝑂superscript𝑡1O(t^{-1}) was proved for infinite horizon Lorentz gases at the semiflow level (after passing to a suspension over a Markov extension and quotienting out stable leaves as in Sections 3 and 6). It was claimed in [25] that this result held also in certain special cases for the Lorentz flow, and that the decay rate O(t(1ϵ))𝑂superscript𝑡1italic-ϵO(t^{-(1-\epsilon)}) held for all ϵ>0italic-ϵ0\epsilon>0 in complete generality. The spurious factor of tϵsuperscript𝑡italic-ϵt^{\epsilon} was then removed in an unpublished preprint “Decay of correlations for flows with unbounded roof function, including the infinite horizon planar periodic Lorentz gas” by the first and third authors. Unfortunately these results for flows do not apply to Lorentz gases since hypothesis (P1) in [25] is not satisfied. The situation is rectified in the current paper. (The unpublished preprint also contained correct results on statistical limit laws such as the central limit theorem for flows with unbounded roof functions. These aspects are completed and extended in [7].)
(b) A drawback of the method in this paper, already present in [18] and inherited by [24, 25, 26], is that at least one of the observables v𝑣v or w𝑤w is required to be Cmsuperscript𝐶𝑚C^{m} in the flow direction. Here m𝑚m can be estimated, with difficulty, but is likely to be quite large. In the case of the infinite horizon Lorentz gas, this excludes certain physically important observables such as velocity. A reasonable project is to attempt to combine methods in this paper with the methods for (stretched) exponential decay in [4, 12] to obtain the decay rate O(t1)𝑂superscript𝑡1O(t^{-1}) for Hölder observables v𝑣v and w𝑤w (cf. the second open question in [26, Section 9]).

In Part I of this paper, we consider results on rapid mixing and polynomial mixing for a class of suspension flows over infinite branch uniformly hyperbolic transformations [30]. In Part II, we show how these results apply to important classes of nonuniformly hyperbolic flows including those mentioned in this introduction. The methods of proof in this paper, especially those in Part I, are fairly straightforward adaptations of those in [26]. The main new contribution of the paper (Section 6 together with Part II) is to develop a general framework whereby large classes of nonuniformly hyperbolic flows, including fundamental examples such as the infinite horizon Lorentz gas, are covered by these methods.

Remark 1.2

The paper has been structured to be as self-contained as possible. It does not seem possible to reduce the results on flows in Part I of this paper to the results on semiflows in [26]. Instead, it is necessary to start from scratch and to emulate, rather than apply directly, the methods in [26]. Some of the more basic estimates in [26] are applicable and are collected together at the beginning of Sections 4 (Lemma 4.1 to Proposition 4.9) and Section 5 (Propositions 5.1 to 5.3), as well as in Section 5.2 (Propositions 5.75.11 and 5.12). Also, results on nonexistence of approximate eigenfunctions in [26] are recalled in Sections 6.2 and Section 8.4.

Notation

We use the “big O𝑂O” and much-less-than\ll notation interchangeably, writing an=O(bn)subscript𝑎𝑛𝑂subscript𝑏𝑛a_{n}=O(b_{n}) or anbnmuch-less-thansubscript𝑎𝑛subscript𝑏𝑛a_{n}\ll b_{n} if there is a constant C>0𝐶0C>0 such that anCbnsubscript𝑎𝑛𝐶subscript𝑏𝑛a_{n}\leq Cb_{n} for all n1𝑛1n\geq 1. There are various “universal” constants C1,,C51subscript𝐶1subscript𝐶51C_{1},\dots,C_{5}\geq 1 depending only on the flow that do not change throughout.

Part I Mixing rates for Gibbs-Markov flows

In this part of the paper, we state and prove results on rapid and polynomial mixing for a class of suspension flows that we call Gibbs-Markov flows. These are suspensions over infinite branch uniformly hyperbolic transformations [30]. In Section 2, we recall material on the noninvertible version, Gibbs-Markov semiflows (suspensions over infinite branch uniformly expanding maps). In Section 3, we consider skew product Gibbs-Markov flows where the roof function is constant along stable leaves and state our main theorems for such flows, namely Theorem 3.1 (rapid mixing) and Theorem 3.2 (polynomial mixing). These are proved in Sections 4 and 5 respectively. In Section 6, we consider an enlarged class of Gibbs-Markov flows that can be reduced to skew products and for which Theorems 3.1 and 3.2 remain valid.

We quickly review notation associated with suspension semiflows and suspension flows. Let (Y,μ)𝑌𝜇(Y,\mu) be a probability space and let F:YY:𝐹𝑌𝑌F:Y\to Y be a measure-preserving transformation. Let φ:Y+:𝜑𝑌superscript\varphi:Y\to{\mathbb{R}}^{+} be an integrable roof function. Define the suspension semiflow/flow

Ft:YφYφ,Yφ={(y,u)Y×[0,):u[0,φ(y)]}/,F_{t}:Y^{\varphi}\to Y^{\varphi},\qquad Y^{\varphi}=\{(y,u)\in Y\times[0,\infty):u\in[0,\varphi(y)]\}/\sim, (1.1)

where (y,φ(y))(Fy,0)similar-to𝑦𝜑𝑦𝐹𝑦0(y,\varphi(y))\sim(Fy,0) and Ft(y,u)=(y,u+t)subscript𝐹𝑡𝑦𝑢𝑦𝑢𝑡F_{t}(y,u)=(y,u+t) computed modulo identifications. An Ftsubscript𝐹𝑡F_{t}-invariant probability measure on Yφsuperscript𝑌𝜑Y^{\varphi} is given by μφ=μ×Lebesgue/Yφ𝑑μsuperscript𝜇𝜑𝜇Lebesguesubscript𝑌𝜑differential-d𝜇\mu^{\varphi}=\mu\times{\rm Lebesgue}/\int_{Y}\varphi\,d\mu.

2 Gibbs-Markov maps and semiflows

In this section, we review definitions and notation from [26, Section 3.1] for a class of Gibbs-Markov semiflows built as suspensions over Gibbs-Markov maps. Standard references for background material on Gibbs-Markov maps are [1, Chapter 4] and [2].

Suppose that (Y¯,μ¯)¯𝑌¯𝜇({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu},\bar{\mu}) is a probability space with an at most countable measurable partition {Y¯j,j1}subscript¯𝑌𝑗𝑗1\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j},\,j\geq 1\} and let F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} be a measure-preserving transformation. For θ(0,1)𝜃01\theta\in(0,1), define dθ(y,y)=θs(y,y)subscript𝑑𝜃𝑦superscript𝑦superscript𝜃𝑠𝑦superscript𝑦d_{\theta}(y,y^{\prime})=\theta^{s(y,y^{\prime})} where the separation time s(y,y)𝑠𝑦superscript𝑦s(y,y^{\prime}) is the least integer n0𝑛0n\geq 0 such that F¯nysuperscript¯𝐹𝑛𝑦\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n}y and F¯nysuperscript¯𝐹𝑛superscript𝑦\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n}y^{\prime} lie in distinct partition elements in {Y¯j}subscript¯𝑌𝑗\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\}. It is assumed that the partition {Y¯j}subscript¯𝑌𝑗\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\} separates trajectories, so s(y,y)=𝑠𝑦superscript𝑦s(y,y^{\prime})=\infty if and only if y=y𝑦superscript𝑦y=y^{\prime}. Then dθsubscript𝑑𝜃d_{\theta} is a metric, called a symbolic metric.

A function v:Y¯:𝑣¯𝑌v:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} is dθsubscript𝑑𝜃d_{\theta}-Lipschitz if |v|θ=supyy|v(y)v(y)|/dθ(y,y)subscript𝑣𝜃subscriptsupremum𝑦superscript𝑦𝑣𝑦𝑣superscript𝑦subscript𝑑𝜃𝑦superscript𝑦|v|_{\theta}=\sup_{y\neq y^{\prime}}|v(y)-v(y^{\prime})|/d_{\theta}(y,y^{\prime}) is finite. Let θ(Y¯)subscript𝜃¯𝑌{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) be the Banach space of Lipschitz functions with norm vθ=|v|+|v|θsubscriptnorm𝑣𝜃subscript𝑣subscript𝑣𝜃\|v\|_{\theta}=|v|_{\infty}+|v|_{\theta}.

More generally (and with a slight abuse of notation), we say that a function v:Y¯:𝑣¯𝑌v:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} is piecewise dθsubscript𝑑𝜃d_{\theta}-Lipschitz if |1Y¯jv|θ=supy,yY¯j,yy|v(y)v(y)|/dθ(y,y)subscriptsubscript1subscript¯𝑌𝑗𝑣𝜃subscriptsupremumformulae-sequence𝑦superscript𝑦subscript¯𝑌𝑗𝑦superscript𝑦𝑣𝑦𝑣superscript𝑦subscript𝑑𝜃𝑦superscript𝑦|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}v|_{\theta}=\sup_{y,y^{\prime}\in{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j},\,y\neq y^{\prime}}|v(y)-v(y^{\prime})|/d_{\theta}(y,y^{\prime}) is finite for all j𝑗j. If in addition, supj|1Y¯jv|θ<subscriptsupremum𝑗subscriptsubscript1subscript¯𝑌𝑗𝑣𝜃\sup_{j}|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}v|_{\theta}<\infty then we say that v𝑣v is uniformly piecewise dθsubscript𝑑𝜃d_{\theta}-Lipschitz. Note that such a function v𝑣v is bounded on partition elements but need not be bounded on Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}.

Definition 2.1

The map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is called a (full branch) Gibbs-Markov map if

  • F¯|Y¯j:Y¯jY¯:evaluated-at¯𝐹subscript¯𝑌𝑗subscript¯𝑌𝑗¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu|_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is a measurable bijection for each j1𝑗1j\geq 1, and

  • The potential function log(dμ¯/dμ¯F¯):Y¯:𝑑¯𝜇𝑑¯𝜇¯𝐹¯𝑌\log(d\bar{\mu}/d\bar{\mu}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu):{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} is uniformly piecewise dθsubscript𝑑𝜃d_{\theta}-Lipschitz for some θ(0,1)𝜃01\theta\in(0,1).

Definition 2.2

A suspension semiflow F¯t:Y¯φY¯φ:subscript¯𝐹𝑡superscript¯𝑌𝜑superscript¯𝑌𝜑\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t}:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\varphi}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\varphi} as in (1.1) is called a Gibbs-Markov semiflow if there exist constants C11subscript𝐶11C_{1}\geq 1, θ(0,1)𝜃01\theta\in(0,1) such that F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is a Gibbs-Markov map, φ:Y¯+:𝜑¯𝑌superscript\varphi:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}}^{+} is an integrable roof function with infφ>0infimum𝜑0\inf\varphi>0, and

|1Y¯jφ|θC1infY¯jφfor all j1.subscriptsubscript1subscript¯𝑌𝑗𝜑𝜃subscript𝐶1subscriptinfimumsubscript¯𝑌𝑗𝜑for all j1|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}\varphi|_{\theta}\leq C_{1}{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}}\varphi\quad\text{for all $j\geq 1$}. (2.1)

(Equivalently, logφ𝜑\log\varphi is uniformly piecewise dθsubscript𝑑𝜃d_{\theta}-Lipschitz.) It follows that supY¯jφ2C1infY¯jφsubscriptsupremumsubscript¯𝑌𝑗𝜑2subscript𝐶1subscriptinfimumsubscript¯𝑌𝑗𝜑{\textstyle\sup_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}}\varphi\leq 2C_{1}{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}}}\varphi for all j1𝑗1j\geq 1.

For b𝑏b\in{\mathbb{R}}, we define the operators

Mb:L(Y¯)L(Y¯),Mbv=eibφvF¯.:subscript𝑀𝑏formulae-sequencesuperscript𝐿¯𝑌superscript𝐿¯𝑌subscript𝑀𝑏𝑣superscript𝑒𝑖𝑏𝜑𝑣¯𝐹M_{b}:L^{\infty}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})\to L^{\infty}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}),\qquad M_{b}v=e^{ib\varphi}v\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu.
Definition 2.3

A subset Z0Y¯subscript𝑍0¯𝑌Z_{0}\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is a finite subsystem of Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} if Z0=n0F¯nZsubscript𝑍0subscript𝑛0superscript¯𝐹𝑛𝑍Z_{0}=\bigcap_{n\geq 0}\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{-n}Z where Z𝑍Z is the union of finitely many elements from the partition {Y¯j}subscript¯𝑌𝑗\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\}. (Note that F¯|Z0:Z0Z0:evaluated-at¯𝐹subscript𝑍0subscript𝑍0subscript𝑍0\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu|_{Z_{0}}:Z_{0}\to Z_{0} is a full one-sided shift on finitely many symbols.)

We say that Mbsubscript𝑀𝑏M_{b} has approximate eigenfunctions on Z0subscript𝑍0Z_{0} if for any α0>0subscript𝛼00\alpha_{0}>0, there exist constants α𝛼\alpha, ξ>α0𝜉subscript𝛼0\xi>\alpha_{0} and C>0𝐶0C>0, and sequences |bk|subscript𝑏𝑘|b_{k}|\to\infty, ψk[0,2π)subscript𝜓𝑘02𝜋\psi_{k}\in[0,2\pi), ukθ(Y¯)subscript𝑢𝑘subscript𝜃¯𝑌u_{k}\in{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) with |uk|1subscript𝑢𝑘1|u_{k}|\equiv 1 and |uk|θC|bk|subscriptsubscript𝑢𝑘𝜃𝐶subscript𝑏𝑘|u_{k}|_{\theta}\leq C|b_{k}|, such that setting nk=[ξln|bk|]subscript𝑛𝑘delimited-[]𝜉subscript𝑏𝑘n_{k}=[\xi\ln|b_{k}|],

|(Mbknkuk)(y)eiψkuk(y)|C|bk|αfor all yZ0k1.superscriptsubscript𝑀subscript𝑏𝑘subscript𝑛𝑘subscript𝑢𝑘𝑦superscript𝑒𝑖subscript𝜓𝑘subscript𝑢𝑘𝑦𝐶superscriptsubscript𝑏𝑘𝛼for all yZ0k1.|(M_{b_{k}}^{n_{k}}u_{k})(y)-e^{i\psi_{k}}u_{k}(y)|\leq C|b_{k}|^{-\alpha}\quad\text{for all $y\in Z_{0}$, $k\geq 1$.} (2.2)
Remark 2.4

For brevity, the statement “Assume absence of approximate eigenfunctions” is the assumption that there exists at least one finite subsystem Z0subscript𝑍0Z_{0} such that Mbsubscript𝑀𝑏M_{b} does not have approximate eigenfunctions on Z0subscript𝑍0Z_{0}.

3 Skew product Gibbs-Markov flows

In this section, we recall the notion of skew product Gibbs-Markov flow [26, Section 4.1] and state our main results on mixing for such flows.

Let (Y,d)𝑌𝑑(Y,d) be a metric space with diamY1diam𝑌1\operatorname{diam}Y\leq 1, and let F:YY:𝐹𝑌𝑌F:Y\to Y be a piecewise continuous map with ergodic F𝐹F-invariant probability measure μ𝜇\mu. Let 𝒲ssuperscript𝒲𝑠{\mathcal{W}}^{s} be a cover of Y𝑌Y by disjoint measurable subsets of Y𝑌Y called stable leaves. For each yY𝑦𝑌y\in Y, let Ws(y)superscript𝑊𝑠𝑦W^{s}(y) denote the stable leaf containing y𝑦y. We require that F(Ws(y))Ws(Fy)𝐹superscript𝑊𝑠𝑦superscript𝑊𝑠𝐹𝑦F(W^{s}(y))\subset W^{s}(Fy) for all yY𝑦𝑌y\in Y.

Let Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} denote the space obtained from Y𝑌Y after quotienting by 𝒲ssuperscript𝒲𝑠{\mathcal{W}}^{s}, with natural projection π¯:YY¯:¯𝜋𝑌¯𝑌\bar{\pi}:Y\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. We assume that the quotient map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is a Gibbs-Markov map as in Definition 2.1, with partition {Y¯j}subscript¯𝑌𝑗\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\}, separation time s(y,y)𝑠𝑦superscript𝑦s(y,y^{\prime}), and ergodic invariant probability measure μ¯=π¯μ¯𝜇subscript¯𝜋𝜇\bar{\mu}=\bar{\pi}_{*}\mu.

Let Yj=π¯1Y¯jsubscript𝑌𝑗superscript¯𝜋1subscript¯𝑌𝑗Y_{j}=\bar{\pi}^{-1}{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}; these form a partition of Y𝑌Y and each Yjsubscript𝑌𝑗Y_{j} is a union of stable leaves. The separation time extends to Y𝑌Y, setting s(y,y)=s(π¯y,π¯y)𝑠𝑦superscript𝑦𝑠¯𝜋𝑦¯𝜋superscript𝑦s(y,y^{\prime})=s(\bar{\pi}y,\bar{\pi}y^{\prime}) for y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y.

Next, we require that there is a measurable subset Y~Y~𝑌𝑌{\widetilde{Y}}\subset Y such that for every yY𝑦𝑌y\in Y there is a unique y~Y~Ws(y)~𝑦~𝑌superscript𝑊𝑠𝑦\tilde{y}\in{\widetilde{Y}}\cap W^{s}(y). Let π:YY~:𝜋𝑌~𝑌\pi:Y\to{\widetilde{Y}} define the associated projection πy=y~𝜋𝑦~𝑦\pi y=\tilde{y}. (Note that Y~~𝑌{\widetilde{Y}} can be identified with Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}, but in general πμμ¯subscript𝜋𝜇¯𝜇\pi_{*}\mu\neq\bar{\mu}.)

We assume that there are constants C21subscript𝐶21C_{2}\geq 1, γ(0,1)𝛾01\gamma\in(0,1) such that for all n0𝑛0n\geq 0,

d(Fny,Fny)𝑑superscript𝐹𝑛𝑦superscript𝐹𝑛superscript𝑦\displaystyle d(F^{n}y,F^{n}y^{\prime}) C2γnabsentsubscript𝐶2superscript𝛾𝑛\displaystyle\leq C_{2}\gamma^{n} for all y,yY with yWs(y),for all y,yY with yWs(y)\displaystyle\quad\text{for all $y,y^{\prime}\in Y$ with $y^{\prime}\in W^{s}(y)$}, (3.1)
d(Fny,Fny)𝑑superscript𝐹𝑛𝑦superscript𝐹𝑛superscript𝑦\displaystyle d(F^{n}y,F^{n}y^{\prime}) C2γs(y,y)nabsentsubscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑛\displaystyle\leq C_{2}\gamma^{s(y,y^{\prime})-n} for all y,yY~𝑦superscript𝑦~𝑌y,y^{\prime}\in{\widetilde{Y}}. (3.2)

Let φ:Y+:𝜑𝑌superscript\varphi:Y\to{\mathbb{R}}^{+} be an integrable roof function with infφ>0infimum𝜑0\inf\varphi>0, and define the suspension flow111Strictly speaking, Ftsubscript𝐹𝑡F_{t} is not always a flow since F𝐹F need not be invertible. However, Ftsubscript𝐹𝑡F_{t} is used as a model for various flows, and it is then a flow when φ𝜑\varphi is the first return to Y𝑌Y, so it is convenient to call it a flow. Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} as in (1.1) with ergodic invariant probability measure μφsuperscript𝜇𝜑\mu^{\varphi}.

In this subsection, we suppose that φ𝜑\varphi is constant along stable leaves and hence projects to a well-defined roof function φ:Y¯+:𝜑¯𝑌superscript\varphi:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}}^{+}. It follows that the suspension flow Ftsubscript𝐹𝑡F_{t} projects to a quotient suspension semiflow F¯t:Y¯φY¯φ:subscript¯𝐹𝑡superscript¯𝑌𝜑superscript¯𝑌𝜑\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t}:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\varphi}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\varphi}. We assume that F¯tsubscript¯𝐹𝑡\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t} is a Gibbs-Markov semiflow (Definition 2.2). In particular, increasing γ(0,1)𝛾01\gamma\in(0,1) if necessary, (2.1) is satisfied in the form

|φ(y)φ(y)|C1infYjφγs(y,y)for all y,yY¯jj1.𝜑𝑦𝜑superscript𝑦subscript𝐶1subscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠𝑦superscript𝑦for all y,yY¯jj1.|\varphi(y)-\varphi(y^{\prime})|\leq C_{1}{\textstyle\inf_{Y_{j}}}\varphi\,\gamma^{s(y,y^{\prime})}\quad\text{for all $y,y^{\prime}\in{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}$, $j\geq 1$.} (3.3)

We call Ftsubscript𝐹𝑡F_{t} a skew product Gibbs-Markov flow, and we say that Ftsubscript𝐹𝑡F_{t} has approximate eigenfunctions if F¯tsubscript¯𝐹𝑡\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t} has approximate eigenfunctions (Definition 2.3).

Fix η(0,1]𝜂01\eta\in(0,1]. For v:Yφ:𝑣superscript𝑌𝜑v:Y^{\varphi}\to{\mathbb{R}}, define

|v|γsubscript𝑣𝛾\displaystyle|v|_{\gamma} =sup(y,u),(y,u)Yφ,yy|v(y,u)v(y,u)|φ(y){d(y,y)+γs(y,y)},absentsubscriptsupremumformulae-sequence𝑦𝑢superscript𝑦𝑢superscript𝑌𝜑𝑦superscript𝑦𝑣𝑦𝑢𝑣superscript𝑦𝑢𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦\displaystyle=\sup_{(y,u),(y^{\prime},u)\in Y^{\varphi},\,y\neq y^{\prime}}\frac{|v(y,u)-v(y^{\prime},u)|}{\varphi(y)\{d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}\}}, vγsubscriptnorm𝑣𝛾\displaystyle\qquad\|v\|_{\gamma} =|v|+|v|γ,absentsubscript𝑣subscript𝑣𝛾\displaystyle=|v|_{\infty}+|v|_{\gamma},
|v|,ηsubscript𝑣𝜂\displaystyle|v|_{\infty,\eta} =sup(y,u),(y,u)Yφ,uu|v(y,u)v(y,u)||uu|η,absentsubscriptsupremumformulae-sequence𝑦𝑢𝑦superscript𝑢superscript𝑌𝜑𝑢superscript𝑢𝑣𝑦𝑢𝑣𝑦superscript𝑢superscript𝑢superscript𝑢𝜂\displaystyle=\sup_{(y,u),(y,u^{\prime})\in Y^{\varphi},\,u\neq u^{\prime}}\frac{|v(y,u)-v(y,u^{\prime})|}{|u-u^{\prime}|^{\eta}}, vγ,ηsubscriptnorm𝑣𝛾𝜂\displaystyle\qquad\|v\|_{\gamma,\eta} =vγ+|v|,η.absentsubscriptnorm𝑣𝛾subscript𝑣𝜂\displaystyle=\|v\|_{\gamma}+|v|_{\infty,\eta}.

(Here |uu|𝑢superscript𝑢|u-u^{\prime}| denotes absolute value, with u,u𝑢superscript𝑢u,u^{\prime} regarded as elements of [0,)0[0,\infty).) Let γ(Yφ)subscript𝛾superscript𝑌𝜑{\mathcal{H}}_{\gamma}(Y^{\varphi}) and γ,η(Yφ)subscript𝛾𝜂superscript𝑌𝜑{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}) be the spaces of observables v:Yφ:𝑣superscript𝑌𝜑v:Y^{\varphi}\to{\mathbb{R}} with vγ<subscriptnorm𝑣𝛾\|v\|_{\gamma}<\infty and vγ,η<subscriptnorm𝑣𝛾𝜂\|v\|_{\gamma,\eta}<\infty respectively.

We say that w:Yφ:𝑤superscript𝑌𝜑w:Y^{\varphi}\to{\mathbb{R}} is differentiable in the flow direction if the limit tw=limt0(wFtw)/tsubscript𝑡𝑤subscript𝑡0𝑤subscript𝐹𝑡𝑤𝑡\partial_{t}w=\lim_{t\to 0}(w\circ F_{t}-w)/t exists pointwise. Note that tw=wusubscript𝑡𝑤𝑤𝑢\partial_{t}w=\frac{\partial w}{\partial u} on the set {(y,u):yY, 0<u<φ(y)}conditional-set𝑦𝑢formulae-sequence𝑦𝑌 0𝑢𝜑𝑦\{(y,u):y\in Y,\,0<u<\varphi(y)\}. Define γ,0,m(Yφ)subscript𝛾0𝑚superscript𝑌𝜑{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi}) to consist of observables w:Yφ:𝑤superscript𝑌𝜑w:Y^{\varphi}\to{\mathbb{R}} that are m𝑚m-times differentiable in the flow direction with derivatives in γ(Yφ)subscript𝛾superscript𝑌𝜑{\mathcal{H}}_{\gamma}(Y^{\varphi}), with norm wγ,0,m=j=0mtjwγsubscriptnorm𝑤𝛾0𝑚superscriptsubscript𝑗0𝑚subscriptnormsuperscriptsubscript𝑡𝑗𝑤𝛾\|w\|_{\gamma,0,m}=\sum_{j=0}^{m}\|\partial_{t}^{j}w\|_{\gamma}.

We can now state the main theoretical results for skew product Gibbs-Markov flows.

Theorem 3.1

Suppose that Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a skew product Gibbs-Markov flow such that φLq(Y)𝜑superscript𝐿𝑞𝑌\varphi\in L^{q}(Y) for all q𝑞q\in{\mathbb{N}}. Assume absence of approximate eigenfunctions.

Then for any q𝑞q\in{\mathbb{N}}, there exists m1𝑚1m\geq 1 and C>0𝐶0C>0 such that

|ρv,w(t)|Cvγwγ,0,mtqfor all vγ(Yφ)wγ,0,m(Yφ)t>1.subscript𝜌𝑣𝑤𝑡𝐶subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾0𝑚superscript𝑡𝑞for all vγ(Yφ)wγ,0,m(Yφ)t>1|\rho_{v,w}(t)|\leq C\|v\|_{\gamma}\|w\|_{\gamma,0,m}\,t^{-q}\quad\text{for all $v\in{\mathcal{H}}_{\gamma}(Y^{\varphi})$, $w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi})$, $t>1$}.
Theorem 3.2

Suppose that Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a skew product Gibbs-Markov flow such that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) for some β>1𝛽1\beta>1. Assume absence of approximate eigenfunctions. Then there exists m1𝑚1m\geq 1 and C>0𝐶0C>0 such that

|ρv,w(t)|Cvγ,ηwγ,0,mt(β1)for all vγ,η(Yφ)wγ,0,m(Yφ)t>1.subscript𝜌𝑣𝑤𝑡𝐶subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾0𝑚superscript𝑡𝛽1for all vγ,η(Yφ)wγ,0,m(Yφ)t>1|\rho_{v,w}(t)|\leq C\|v\|_{\gamma,\eta}\|w\|_{\gamma,0,m}\,t^{-(\beta-1)}\quad\text{for all $v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi})$, $w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi})$, $t>1$}.
Remark 3.3

Our result on polynomial mixing, Theorem 3.2, implies the result on rapid mixing, Theorem 3.1 (for a slightly more restricted class of observables). However, the proof of Theorem 3.1 plays a crucial role in the proof of Theorem 3.2, justifying the movement of certain contours of integration to the imaginary axis after the truncation step in Section 5.2. Hence, it is not possible to bypass Theorem 3.1 even when only polynomial mixing is of interest.

These results are proved in Sections 4 and 5 respectively. For future reference, we mention the following estimates. Define φn=j=0n1φFjsubscript𝜑𝑛superscriptsubscript𝑗0𝑛1𝜑superscript𝐹𝑗\varphi_{n}=\sum_{j=0}^{n-1}\varphi\circ F^{j}.

Proposition 3.4

Let η(0,β)𝜂0𝛽\eta\in(0,\beta). Then

(a) YφηFi1{φn>t}𝑑μ(n+1)Yφη1{φ>t/n}𝑑μsubscript𝑌superscript𝜑𝜂superscript𝐹𝑖subscript1subscript𝜑𝑛𝑡differential-d𝜇𝑛1subscript𝑌superscript𝜑𝜂subscript1𝜑𝑡𝑛differential-d𝜇\int_{Y}\varphi^{\eta}\circ F^{i}1_{\{\varphi_{n}>t\}}\,d\mu\leq(n+1)\int_{Y}\varphi^{\eta}1_{\{\varphi>t/n\}}\,d\mu for all i0𝑖0i\geq 0, n1𝑛1n\geq 1, t>0𝑡0t>0.

(b) If μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) for some β>1𝛽1\beta>1, then Yφη1{φ>t}𝑑μ=O(t(βη))subscript𝑌superscript𝜑𝜂subscript1𝜑𝑡differential-d𝜇𝑂superscript𝑡𝛽𝜂\int_{Y}\varphi^{\eta}1_{\{\varphi>t\}}\,d\mu=O(t^{-(\beta-\eta)}).

Proof.

Writing φηFi=φηFi1{φFi>t/n}+φηFi1{φFit/n}superscript𝜑𝜂superscript𝐹𝑖superscript𝜑𝜂superscript𝐹𝑖subscript1𝜑superscript𝐹𝑖𝑡𝑛superscript𝜑𝜂superscript𝐹𝑖subscript1𝜑superscript𝐹𝑖𝑡𝑛\varphi^{\eta}\circ F^{i}=\varphi^{\eta}\circ F^{i}1_{\{\varphi\circ F^{i}>t/n\}}+\varphi^{\eta}\circ F^{i}1_{\{\varphi\circ F^{i}\leq t/n\}}, we compute that

Yφηsubscript𝑌superscript𝜑𝜂\displaystyle\int_{Y}\varphi^{\eta} Fi1{φn>t}dμabsentsuperscript𝐹𝑖subscript1subscript𝜑𝑛𝑡𝑑𝜇\displaystyle\circ F^{i}1_{\{\varphi_{n}>t\}}\,d\mu
=YφηFi1{φFi>t/n}1{φn>t}𝑑μ+YφηFi1{φFit/n}1{φn>t}𝑑μabsentsubscript𝑌superscript𝜑𝜂superscript𝐹𝑖subscript1𝜑superscript𝐹𝑖𝑡𝑛subscript1subscript𝜑𝑛𝑡differential-d𝜇subscript𝑌superscript𝜑𝜂superscript𝐹𝑖subscript1𝜑superscript𝐹𝑖𝑡𝑛subscript1subscript𝜑𝑛𝑡differential-d𝜇\displaystyle=\int_{Y}\varphi^{\eta}\circ F^{i}1_{\{\varphi\circ F^{i}>t/n\}}1_{\{\varphi_{n}>t\}}\,d\mu+\int_{Y}\varphi^{\eta}\circ F^{i}1_{\{\varphi\circ F^{i}\leq t/n\}}1_{\{\varphi_{n}>t\}}\,d\mu
YφηFi1{φFi>t/n}𝑑μ+j=0n1Y(tn)η1{φFj>t/n}𝑑μabsentsubscript𝑌superscript𝜑𝜂superscript𝐹𝑖subscript1𝜑superscript𝐹𝑖𝑡𝑛differential-d𝜇superscriptsubscript𝑗0𝑛1subscript𝑌superscript𝑡𝑛𝜂subscript1𝜑superscript𝐹𝑗𝑡𝑛differential-d𝜇\displaystyle\leq\int_{Y}\varphi^{\eta}\circ F^{i}1_{\{\varphi\circ F^{i}>t/n\}}\,d\mu+\sum_{j=0}^{n-1}\int_{Y}\Big{(}\frac{t}{n}\Big{)}^{\eta}1_{\{\varphi\circ F^{j}>t/n\}}\,d\mu
=Yφη1{φ>t/n}𝑑μ+nY(tn)η1{φ>t/n}𝑑μ(n+1)Yφη1{φ>t/n}𝑑μ,absentsubscript𝑌superscript𝜑𝜂subscript1𝜑𝑡𝑛differential-d𝜇𝑛subscript𝑌superscript𝑡𝑛𝜂subscript1𝜑𝑡𝑛differential-d𝜇𝑛1subscript𝑌superscript𝜑𝜂subscript1𝜑𝑡𝑛differential-d𝜇\displaystyle=\int_{Y}\varphi^{\eta}1_{\{\varphi>t/n\}}\,d\mu+n\int_{Y}\Big{(}\frac{t}{n}\Big{)}^{\eta}1_{\{\varphi>t/n\}}\,d\mu\leq(n+1)\int_{Y}\varphi^{\eta}1_{\{\varphi>t/n\}}\,d\mu,

proving part (a). Part (b) is standard (see for example [26, Proposition 8.5]). ∎

4 Rapid mixing for skew product Gibbs-Markov flows

In this section, we consider skew product Gibbs-Markov flows Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} for which the roof function φ:Y+:𝜑𝑌superscript\varphi:Y\to{\mathbb{R}}^{+} lies in Lq(Y)superscript𝐿𝑞𝑌L^{q}(Y) for all q1𝑞1q\geq 1. For such flows, we prove Theorem 3.1, namely that absence of approximate eigenfunctions is a sufficient condition for rapid mixing. For notational convenience, we suppose that infφ1infimum𝜑1\inf\varphi\geq 1.

4.1 Some notation and results from [26]

Let ={s:Res>0}conditional-set𝑠Re𝑠0{\mathbb{H}}=\{s\in{\mathbb{C}}:\operatorname{Re}s>0\} and ¯={s:Res0}¯conditional-set𝑠Re𝑠0{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}=\{s\in{\mathbb{C}}:\operatorname{Re}s\geq 0\}. The Laplace transform ρ^v,w(s)=0estρv,w(t)𝑑tsubscript^𝜌𝑣𝑤𝑠superscriptsubscript0superscript𝑒𝑠𝑡subscript𝜌𝑣𝑤𝑡differential-d𝑡\hat{\rho}_{v,w}(s)=\int_{0}^{\infty}e^{-st}\rho_{v,w}(t)\,dt of the correlation function ρv,wsubscript𝜌𝑣𝑤\rho_{v,w} is analytic on {\mathbb{H}}.

Lemma 4.1 ( [26, Lemma 6.2] )

Let vL1(Yφ)𝑣superscript𝐿1superscript𝑌𝜑v\in L^{1}(Y^{\varphi}), ϵ>0italic-ϵ0\epsilon>0, r1𝑟1r\geq 1. Suppose that

  • (i)

    sρ^v,w(s)maps-to𝑠subscript^𝜌𝑣𝑤𝑠s\mapsto\hat{\rho}_{v,w}(s) is continuous on {Res[0,ϵ]}Re𝑠0italic-ϵ\{\operatorname{Re}s\in[0,\epsilon]\} and bρ^v,w(ib)maps-to𝑏subscript^𝜌𝑣𝑤𝑖𝑏b\mapsto\hat{\rho}_{v,w}(ib) is Crsuperscript𝐶𝑟C^{r} on {\mathbb{R}} for all wγ(Yφ)𝑤subscript𝛾superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}).

  • (ii)

    There exist constants C,α>0𝐶𝛼0C,\alpha>0 such that

    |ρ^v,w(s)|C(|b|+1)αwγand|ρ^v,w(j)(ib)|C(|b|+1)αwγ,formulae-sequencesubscript^𝜌𝑣𝑤𝑠𝐶superscript𝑏1𝛼subscriptnorm𝑤𝛾andsuperscriptsubscript^𝜌𝑣𝑤𝑗𝑖𝑏𝐶superscript𝑏1𝛼subscriptnorm𝑤𝛾|\hat{\rho}_{v,w}(s)|\leq C(|b|+1)^{\alpha}\|w\|_{\gamma}\quad\text{and}\quad|\hat{\rho}_{v,w}^{(j)}(ib)|\leq C(|b|+1)^{\alpha}\|w\|_{\gamma},

    for all wγ(Yφ)𝑤subscript𝛾superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}), jr𝑗𝑟j\leq r, and all s=a+ib𝑠𝑎𝑖𝑏s=a+ib\in{\mathbb{C}} with a[0,ϵ]𝑎0italic-ϵa\in[0,\epsilon].

Let m=α+2𝑚𝛼2m=\lceil\alpha\rceil+2. Then there exists a constant C>0superscript𝐶0C^{\prime}>0 depending only on r𝑟r and α𝛼\alpha, such that

|ρv,w(t)|CCwγ,0,mtrfor all wγ,0,m(Yφ)t>1.subscript𝜌𝑣𝑤𝑡𝐶superscript𝐶subscriptnorm𝑤𝛾0𝑚superscript𝑡𝑟for all wγ,0,m(Yφ)t>1|\rho_{v,w}(t)|\leq CC^{\prime}\|w\|_{\gamma,0,m}\,t^{-r}\quad\text{for all $w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi})$, $t>1$}.

Remark 4.2

Since ρ^v,wsubscript^𝜌𝑣𝑤\hat{\rho}_{v,w} is not a priori well-defined on ¯¯{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}, the conditions in this lemma should be interpreted in the usual way, namely that ρ^v,w::subscript^𝜌𝑣𝑤\hat{\rho}_{v,w}:{\mathbb{H}}\to{\mathbb{C}} extends to a function g:¯:𝑔¯g:{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{C}} satisfying the desired conditions (i) and (ii). The conclusion for ρv,wsubscript𝜌𝑣𝑤\rho_{v,w} then follows from a standard uniqueness argument.

For completeness, we provide the uniqueness argument. By [26, Corollary 6.1], the inverse Laplace transform of ρ^v,wsubscript^𝜌𝑣𝑤\hat{\rho}_{v,w} can be computed by integrating along a contour in {\mathbb{H}}. Since gρ^v,w𝑔subscript^𝜌𝑣𝑤g\equiv\hat{\rho}_{v,w} on {\mathbb{H}}, we can compute the inverse Laplace transform f𝑓f of g𝑔g using the same contour, and we obtain ρv,wfsubscript𝜌𝑣𝑤𝑓\rho_{v,w}\equiv f. Hence ρ^v,wgsubscript^𝜌𝑣𝑤𝑔\hat{\rho}_{v,w}\equiv g is well-defined on ¯¯{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu} and satisfies conditions (i) and (ii), so the conclusion follows from [26, Lemma 6.2].

Define vs(y)=0φ(y)esuv(y,u)𝑑usubscript𝑣𝑠𝑦superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝑢𝑣𝑦𝑢differential-d𝑢v_{s}(y)=\int_{0}^{\varphi(y)}e^{su}v(y,u)\,du and w^(s)(y)=0φ(y)esuw(y,u)𝑑u^𝑤𝑠𝑦superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝑢𝑤𝑦𝑢differential-d𝑢{\widehat{w}}(s)(y)=\int_{0}^{\varphi(y)}e^{-su}w(y,u)\,du.

Proposition 4.3 ( [26, Proposition 6.3 and Corollary 8.6] )

Let v,wL(Yφ)𝑣𝑤superscript𝐿superscript𝑌𝜑v,\,w\in L^{\infty}(Y^{\varphi}) with Yφv𝑑μφ=0subscriptsuperscript𝑌𝜑𝑣differential-dsuperscript𝜇𝜑0\int_{Y^{\varphi}}v\,d\mu^{\varphi}=0. Then ρ^v,w=n=0J^nsubscript^𝜌𝑣𝑤superscriptsubscript𝑛0subscript^𝐽𝑛\hat{\rho}_{v,w}=\sum_{n=0}^{\infty}{\widehat{J}}_{n} on {\mathbb{H}} where J^nsubscript^𝐽𝑛{\widehat{J}}_{n} is the Laplace transform of an Lsuperscript𝐿L^{\infty} function Jn:[0,):subscript𝐽𝑛0J_{n}:[0,\infty)\to{\mathbb{R}} for n0𝑛0n\geq 0, and

J^n(s)=|φ|11Yesφnvsw^(s)Fn𝑑μfor all s¯n1.subscript^𝐽𝑛𝑠superscriptsubscript𝜑11subscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛differential-d𝜇for all s¯n1.{\widehat{J}}_{n}(s)=|\varphi|_{1}^{-1}{\textstyle\int}_{Y}e^{-s\varphi_{n}}v_{s}\,{\widehat{w}}(s)\circ F^{n}\,d\mu\qquad\text{for all $s\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}$, $n\geq 1$.}

Moreover, |J0(t)|=O(|v||w|t(β1))subscript𝐽0𝑡𝑂subscript𝑣subscript𝑤superscript𝑡𝛽1|J_{0}(t)|=O(|v|_{\infty}|w|_{\infty}\,t^{-(\beta-1)}).222 All series that we consider on {\mathbb{H}} are absolutely convergent for elementary reasons. Details are given in Lemma 4.11 but are generally omitted.

Let R:L1(Y¯)L1(Y¯):𝑅superscript𝐿1¯𝑌superscript𝐿1¯𝑌R:L^{1}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})\to L^{1}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) denote the transfer operator corresponding to the Gibbs-Markov quotient map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. So Y¯vwF¯𝑑μ¯=Y¯Rvw𝑑μ¯subscript¯𝑌𝑣𝑤¯𝐹differential-d¯𝜇subscript¯𝑌𝑅𝑣𝑤differential-d¯𝜇\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}v\,w\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu\,d\bar{\mu}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}Rv\,w\,d\bar{\mu} for all vL1(Y¯)𝑣superscript𝐿1¯𝑌v\in L^{1}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) and wL(Y¯)𝑤superscript𝐿¯𝑌w\in L^{\infty}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}). Also, for s¯𝑠¯s\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}, define the twisted transfer operators

R^(s):L1(Y¯)L1(Y¯),R^(s)v=R(esφv).:^𝑅𝑠formulae-sequencesuperscript𝐿1¯𝑌superscript𝐿1¯𝑌^𝑅𝑠𝑣𝑅superscript𝑒𝑠𝜑𝑣{\widehat{R}}(s):L^{1}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})\to L^{1}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}),\qquad{\widehat{R}}(s)v=R(e^{-s\varphi}v).
Proposition 4.4

Let θ(0,1)𝜃01\theta\in(0,1) be as in Definition 2.1. There is a constant C>0𝐶0C>0 such that

RnvθCdμ¯(d)1dvθfor vθ(Y¯)n1,subscriptnormsuperscript𝑅𝑛𝑣𝜃𝐶subscript𝑑¯𝜇𝑑subscriptnormsubscript1𝑑𝑣𝜃for vθ(Y¯)n1\textstyle\|R^{n}v\|_{\theta}\leq C\sum_{d}\bar{\mu}(d)\|1_{d}v\|_{\theta}\quad\text{for $v\in{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})$, $n\geq 1$},

where the sum is over n𝑛n-cylinders d=i=0,,n1F¯iY¯ji𝑑subscript𝑖0𝑛1superscript¯𝐹𝑖subscript¯𝑌subscript𝑗𝑖d=\bigcap_{i=0,\dots,n-1}\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{-i}{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j_{i}}, j0,,jn11subscript𝑗0subscript𝑗𝑛11j_{0},\dots,j_{n-1}\geq 1.

Proof.

This follows from [26, Corollary 7.2]. ∎

For the remainder of this subsection, we suppose that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) where β>1𝛽1\beta>1. Fix q>0𝑞0q>0 with

max{1,β1}<q<β.1𝛽1𝑞𝛽\max\{1,\beta-1\}<q<\beta.

Let η(0,1]𝜂01\eta\in(0,1], γ(0,1)𝛾01\gamma\in(0,1) are as in Section 3. Shrinking η𝜂\eta if needed, we may suppose without loss that

q+2η<β,𝑞2𝜂𝛽q+2\eta<\beta,

Let γ1=γηsubscript𝛾1superscript𝛾𝜂\gamma_{1}=\gamma^{\eta} and increase θ𝜃\theta if needed so that θ[γ11/3,1)𝜃superscriptsubscript𝛾1131\theta\in[\gamma_{1}^{1/3},1).

A function f::𝑓f:{\mathbb{R}}\to{\mathbb{R}} is said to be Cqsuperscript𝐶𝑞C^{q} if f𝑓f is C[q]superscript𝐶delimited-[]𝑞C^{[q]} and f([q])superscript𝑓delimited-[]𝑞f^{([q])} is (q[q])𝑞delimited-[]𝑞(q-[q])-Hölder. Moreover, given g:[0,):𝑔0g:{\mathbb{R}}\to[0,\infty) and E𝐸E\subset{\mathbb{R}}, we write |f(q)(b)|g(b)superscript𝑓𝑞𝑏𝑔𝑏|f^{(q)}(b)|\leq g(b) for bE𝑏𝐸b\in E if for all b,bE𝑏superscript𝑏𝐸b,b^{\prime}\in E,

|f(k)(b)|g(b),k=0,1,,[q],and|f([q])(b)f([q])(b)|(g(b)+g(b))|bb|q[q].formulae-sequencesuperscript𝑓𝑘𝑏𝑔𝑏formulae-sequence𝑘01delimited-[]𝑞andsuperscript𝑓delimited-[]𝑞𝑏superscript𝑓delimited-[]𝑞superscript𝑏𝑔𝑏𝑔superscript𝑏superscript𝑏superscript𝑏𝑞delimited-[]𝑞|f^{(k)}(b)|\leq g(b),\;k=0,1,\dots,[q],\;\text{and}\;|f^{([q])}(b)-f^{([q])}(b^{\prime})|\leq(g(b)+g(b^{\prime}))|b-b^{\prime}|^{q-[q]}.

For f:¯:𝑓¯f:{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} and E¯𝐸¯E\subset{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}, we write |f(q)(s)|g(s)superscript𝑓𝑞𝑠𝑔𝑠|f^{(q)}(s)|\leq g(s) for sE𝑠𝐸s\in E if |f(q)(ib)|g(b)superscript𝑓𝑞𝑖𝑏𝑔𝑏|f^{(q)}(ib)|\leq g(b) in the sense just given for ibE𝑖𝑏𝐸ib\in E and |f(k)(s)|g(s)superscript𝑓𝑘𝑠𝑔𝑠|f^{(k)}(s)|\leq g(s) for sE𝑠𝐸s\in E, k=0,,[q]𝑘0delimited-[]𝑞k=0,\dots,[q]. The same conventions apply to operator-valued functions on ¯¯{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}.

Remark 4.5

Restricting to q𝑞q as above enables us to obtain estimates for the rapid mixing and polynomially mixing situations simultaneously hence avoiding a certain amount of repetition. The trade off is that the proof of Theorem 3.1 is considerably more difficult. The reader interested only in the rapid mixing case can restrict to integer values of q𝑞q with greatly simplified arguments [26, Section 7] (also see version 3 of our preprint on arxiv).

Following [26, Section 7.4], there exist constants M0,M1subscript𝑀0subscript𝑀1M_{0},\,M_{1} and a scale of equivalent norms

vb=max{|v|,|v|θM0(|b|+1)},b,formulae-sequencesubscriptnorm𝑣𝑏subscript𝑣subscript𝑣𝜃subscript𝑀0𝑏1𝑏\|v\|_{b}=\max\Big{\{}|v|_{\infty},\,\frac{|v|_{\theta}}{M_{0}(|b|+1)}\Big{\}},\quad b\in{\mathbb{R}},

on θ(Y¯)subscript𝜃¯𝑌{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) such that

R^(s)nbM1for all s=a+ib with a[0,1] and all n1.subscriptnorm^𝑅superscript𝑠𝑛𝑏subscript𝑀1for all s=a+ib with a[0,1] and all n1\|{\widehat{R}}(s)^{n}\|_{b}\leq M_{1}\quad\text{for all $s=a+ib\in{\mathbb{C}}$ with $a\in[0,1]$ and all $n\geq 1$}. (4.1)
Proposition 4.6

There is a constant C>0𝐶0C>0 such that

R^(q)(s)bCfor all s=a+ib with 0a1.subscriptnormsuperscript^𝑅𝑞𝑠𝑏𝐶for all s=a+ib with 0a1.\textstyle\|{\widehat{R}}^{(q)}(s)\|_{b}\leq C\quad\text{for all $s=a+ib\in{\mathbb{C}}$ with $0\leq a\leq 1$.}
Proof.

It is shown in [26, Proposition 8.7] that R^(q)(s)θC(|b|+1)subscriptnormsuperscript^𝑅𝑞𝑠𝜃𝐶𝑏1\|{\widehat{R}}^{(q)}(s)\|_{\theta}\leq C(|b|+1). Using the definition of b\|\;\|_{b}, the desired estimate follows by exactly the same argument. ∎

Remark 4.7

Estimates such as those for R^(q)superscript^𝑅𝑞{\widehat{R}}^{(q)} in Proposition 4.6 hold equally for R^(q)superscript^𝑅superscript𝑞{\widehat{R}}^{(q^{\prime})} for all q<qsuperscript𝑞𝑞q^{\prime}<q. We use this observation without comment throughout.

Define δ=¯Bδ(0)subscript𝛿¯subscript𝐵𝛿0{\mathbb{H}}_{\delta}={\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\cap B_{\delta}(0) for δ>0𝛿0\delta>0. Let T^=(IR^)1^𝑇superscript𝐼^𝑅1{\widehat{T}}=(I-{\widehat{R}})^{-1}. We have the key Dolgopyat estimate:

Proposition 4.8

Assume absence of approximate eigenfunctions. Then T^(s):θ(Y¯)θ(Y¯):^𝑇𝑠subscript𝜃¯𝑌subscript𝜃¯𝑌{\widehat{T}}(s):{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})\to{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) is a well-defined bounded operator for s¯{0}𝑠¯0s\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\setminus\{0\}. Moreover, for any δ>0𝛿0\delta>0, there exists α,C>0𝛼𝐶0\alpha,\,C>0 such that

T^(q)(s)θC|b|αfor all s=a+ibδ with 0a1.subscriptnormsuperscript^𝑇𝑞𝑠𝜃𝐶superscript𝑏𝛼for all s=a+ibδ with 0a1.\|{\widehat{T}}^{(q)}(s)\|_{\theta}\leq C|b|^{\alpha}\quad\text{for all $s=a+ib\in{\mathbb{C}}\setminus{\mathbb{H}}_{\delta}$ with $0\leq a\leq 1$.}

Proof.

For the region 0a10𝑎10\leq a\leq 1, |b|δ𝑏𝛿|b|\geq\delta, this is explicit in [26, Corollary 8.10]. The remaining region A=([0,1]×[δ,δ])δ𝐴01𝛿𝛿subscript𝛿A=([0,1]\times[-\delta,\delta])\setminus{\mathbb{H}}_{\delta} is bounded. Also, 1specR^(s)1spec^𝑅𝑠1\not\in\operatorname{spec}{\widehat{R}}(s) for s¯{0}𝑠¯0s\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\setminus\{0\} by [26, Proposition 7.8(b) and Theorem 7.10(a)]. Hence T^(q)θsubscriptnormsuperscript^𝑇𝑞𝜃\|{\widehat{T}}^{(q)}\|_{\theta} is bounded on A𝐴A by Proposition 4.6.  ∎

Proposition 4.9 ( [26, Proposition 7.8 and Corollary 7.9] )

There exists δ>0𝛿0\delta>0 such that R^(s):θ(Y¯)θ(Y¯):^𝑅𝑠subscript𝜃¯𝑌subscript𝜃¯𝑌{\widehat{R}}(s):{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu})\to{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) has a Cqsuperscript𝐶𝑞C^{q} family of simple eigenvalues λ(s)𝜆𝑠\lambda(s), sδ𝑠subscript𝛿s\in{\mathbb{H}}_{\delta}, isolated in specR^(s)spec^𝑅𝑠\operatorname{spec}{\widehat{R}}(s), with λ(0)=1𝜆01\lambda(0)=1, λ(0)=|φ|1superscript𝜆0subscript𝜑1\lambda^{\prime}(0)=-|\varphi|_{1}, |λ(s)|1𝜆𝑠1|\lambda(s)|\leq 1. The corresponding spectral projections P(s)𝑃𝑠P(s) form a Cqsuperscript𝐶𝑞C^{q} family of operators on θ(Y¯)subscript𝜃¯𝑌{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) with P(0)v=Y¯v𝑑μ¯𝑃0𝑣subscript¯𝑌𝑣differential-d¯𝜇P(0)v=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}v\,d\bar{\mu}. ∎

4.2 Approximation of vssubscript𝑣𝑠v_{s} and w^(s)^𝑤𝑠{\widehat{w}}(s)

The first step is to approximate vs,w^(s):Y:subscript𝑣𝑠^𝑤𝑠𝑌v_{s},\,{\widehat{w}}(s):Y\to{\mathbb{C}} by functions that are constant on stable leaves and hence well-defined on Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}.

For k0𝑘0k\geq 0, define Δk:L(Y)L(Y):subscriptΔ𝑘superscript𝐿𝑌superscript𝐿𝑌\Delta_{k}:L^{\infty}(Y)\to L^{\infty}(Y),

Δkw=wFkπwFk1πF,k1,Δ0w=wπ.formulae-sequencesubscriptΔ𝑘𝑤𝑤superscript𝐹𝑘𝜋𝑤superscript𝐹𝑘1𝜋𝐹formulae-sequence𝑘1subscriptΔ0𝑤𝑤𝜋\Delta_{k}w=w\circ F^{k}\circ\pi-w\circ F^{k-1}\circ\pi\circ F,\;k\geq 1,\quad\Delta_{0}w=w\circ\pi.
Proposition 4.10

Let wL(Y)𝑤superscript𝐿𝑌w\in L^{\infty}(Y). Then

  • (a)

    ΔkwsubscriptΔ𝑘𝑤\Delta_{k}w is constant along stable leaves.

  • (b)

    k=0n(Δkw)Fnk=wFnπsuperscriptsubscript𝑘0𝑛subscriptΔ𝑘𝑤superscript𝐹𝑛𝑘𝑤superscript𝐹𝑛𝜋\sum_{k=0}^{n}(\Delta_{k}w)\circ F^{n-k}=w\circ F^{n}\circ\pi.

Proof.

Part (a) is immediate from the definition and part (b) follows by induction.  ∎

Define

V^j(s)=esφFjΔjvs,W^k(s)=esφkΔkw^(s).formulae-sequencesubscript^𝑉𝑗𝑠superscript𝑒𝑠𝜑superscript𝐹𝑗subscriptΔ𝑗subscript𝑣𝑠subscript^𝑊𝑘𝑠superscript𝑒𝑠subscript𝜑𝑘subscriptΔ𝑘^𝑤𝑠{\widehat{V}}_{j}(s)=e^{-s\varphi\circ F^{j}}\Delta_{j}v_{s},\qquad{\widehat{W}}_{k}(s)=e^{-s\varphi_{k}}\Delta_{k}{\widehat{w}}(s).

By Proposition 4.10(a), these can be regarded as functions V¯jsubscript¯𝑉𝑗{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}, W¯ksubscript¯𝑊𝑘{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k} on Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. Similarly we write Δkw¯L(Y¯)¯subscriptΔ𝑘𝑤superscript𝐿¯𝑌\overline{\Delta_{k}w}\in L^{\infty}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}).

Also, for k0𝑘0k\geq 0, we define Ek:L(Y)L(Y):subscript𝐸𝑘superscript𝐿𝑌superscript𝐿𝑌E_{k}:L^{\infty}(Y)\to L^{\infty}(Y),

Ekw=wFkwFkπ.subscript𝐸𝑘𝑤𝑤superscript𝐹𝑘𝑤superscript𝐹𝑘𝜋E_{k}w=w\circ F^{k}-w\circ F^{k}\circ\pi.
Lemma 4.11

Let v,wL(Yφ)𝑣𝑤superscript𝐿superscript𝑌𝜑v,w\in L^{\infty}(Y^{\varphi}). Then

ρ^v,w=J^0+|φ|11(n=1A^n+n=1k=0n1B^n,k+j=0k=0C^j,k),subscript^𝜌𝑣𝑤subscript^𝐽0superscriptsubscript𝜑11superscriptsubscript𝑛1subscript^𝐴𝑛superscriptsubscript𝑛1superscriptsubscript𝑘0𝑛1subscript^𝐵𝑛𝑘superscriptsubscript𝑗0superscriptsubscript𝑘0subscript^𝐶𝑗𝑘\hat{\rho}_{v,w}={\widehat{J}}_{0}+|\varphi|_{1}^{-1}\Big{(}\sum_{n=1}^{\infty}{\widehat{A}}_{n}+\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}{\widehat{B}}_{n,k}+\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}{\widehat{C}}_{j,k}\Big{)},

on {\mathbb{H}}, where

A^n(s)subscript^𝐴𝑛𝑠\displaystyle{\widehat{A}}_{n}(s) =Yesφnvs(En1w^(s))F𝑑μ,absentsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠subscript𝐸𝑛1^𝑤𝑠𝐹differential-d𝜇\displaystyle=\int_{Y}e^{-s\varphi_{n}}v_{s}\,(E_{n-1}{\widehat{w}}(s))\circ F\,d\mu,
B^n,k(s)subscript^𝐵𝑛𝑘𝑠\displaystyle{\widehat{B}}_{n,k}(s) =YesφnFnEnvs(Δkw^(s))F2nk𝑑μ,absentsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛superscript𝐹𝑛subscript𝐸𝑛subscript𝑣𝑠subscriptΔ𝑘^𝑤𝑠superscript𝐹2𝑛𝑘differential-d𝜇\displaystyle=\int_{Y}e^{-s\varphi_{n}\circ F^{n}}E_{n}v_{s}\;(\Delta_{k}{\widehat{w}}(s))\circ F^{2n-k}\,d\mu,
C^j,k(s)subscript^𝐶𝑗𝑘𝑠\displaystyle{\widehat{C}}_{j,k}(s) =Y¯R^(s)max{jk1,0}T^(s)Rj+1V¯j(s)W¯k(s)𝑑μ¯.absentsubscript¯𝑌^𝑅superscript𝑠𝑗𝑘10^𝑇𝑠superscript𝑅𝑗1subscript¯𝑉𝑗𝑠subscript¯𝑊𝑘𝑠differential-d¯𝜇\displaystyle=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\widehat{R}}(s)^{\max\{j-k-1,0\}}{\widehat{T}}(s)R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(s)\;{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(s)\,d\bar{\mu}.

All of these series are absolutely convergent exponentially quickly, pointwise on {\mathbb{H}}.

Proof.

Since this result is set in the right-half complex plane, the final statement is elementary. We sketch the arguments. Let s𝑠s\in{\mathbb{C}} with a=Res>0𝑎Re𝑠0a=\operatorname{Re}s>0. It is clear that |vs|a1|v|eaφsubscript𝑣𝑠superscript𝑎1subscript𝑣superscript𝑒𝑎𝜑|v_{s}|\leq a^{-1}|v|_{\infty}e^{a\varphi} and |w^(s)|a1|w|subscript^𝑤𝑠superscript𝑎1subscript𝑤|{\widehat{w}}(s)|_{\infty}\leq a^{-1}|w|_{\infty}. Hence |A^n(s)|2a2|v||w|ea(n1)subscript^𝐴𝑛𝑠2superscript𝑎2subscript𝑣subscript𝑤superscript𝑒𝑎𝑛1|{\widehat{A}}_{n}(s)|\leq 2a^{-2}|v|_{\infty}|w|_{\infty}e^{-a(n-1)} and |B^n,k(s)|4a2|v||w|ea(n1)subscript^𝐵𝑛𝑘𝑠4superscript𝑎2subscript𝑣subscript𝑤superscript𝑒𝑎𝑛1|{\widehat{B}}_{n,k}(s)|\leq 4a^{-2}|v|_{\infty}|w|_{\infty}e^{-a(n-1)}. Similarly, |V^j(s)|2a1|v|subscriptsubscript^𝑉𝑗𝑠2superscript𝑎1subscript𝑣|{\widehat{V}}_{j}(s)|_{\infty}\leq 2a^{-1}|v|_{\infty} and |W^k(s)|2a1|w|eaksubscriptsubscript^𝑊𝑘𝑠2superscript𝑎1subscript𝑤superscript𝑒𝑎𝑘|{\widehat{W}}_{k}(s)|_{\infty}\leq 2a^{-1}|w|_{\infty}e^{-ak}. As an operator on L(Y)superscript𝐿𝑌L^{\infty}(Y), we have |R^(s)|easubscript^𝑅𝑠superscript𝑒𝑎|{\widehat{R}}(s)|_{\infty}\leq e^{-a}. Hence |C^j,k(s)|4a2(1ea)1|v||w|eamax(j1,k)subscript^𝐶𝑗𝑘𝑠4superscript𝑎2superscript1superscript𝑒𝑎1subscript𝑣subscript𝑤superscript𝑒𝑎𝑗1𝑘|{\widehat{C}}_{j,k}(s)|\leq 4a^{-2}(1-e^{-a})^{-1}|v|_{\infty}|w|_{\infty}e^{-a\max(j-1,k)}.

By Proposition 4.3, ρ^v,w(s)=J^0(s)+|φ|11n=1Yesφnvsw^(s)Fn𝑑μsubscript^𝜌𝑣𝑤𝑠subscript^𝐽0𝑠superscriptsubscript𝜑11superscriptsubscript𝑛1subscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛differential-d𝜇\hat{\rho}_{v,w}(s)={\widehat{J}}_{0}(s)+|\varphi|_{1}^{-1}\sum_{n=1}^{\infty}\int_{Y}e^{-s\varphi_{n}}v_{s}\;{\widehat{w}}(s)\circ F^{n}\,d\mu for s𝑠s\in{\mathbb{H}}. By Proposition 4.10(b), for each n1𝑛1n\geq 1,

Yesφnvsw^(s)Fn𝑑μsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛differential-d𝜇\displaystyle\int_{Y}e^{-s\varphi_{n}}v_{s}\;{\widehat{w}}(s)\circ F^{n}\,d\mu =Yesφnvsw^(s)Fn1πF𝑑μabsentsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛1𝜋𝐹differential-d𝜇\displaystyle=\int_{Y}e^{-s\varphi_{n}}v_{s}\;{\widehat{w}}(s)\circ F^{n-1}\circ\pi\circ F\,d\mu
+Yesφnvs(w^(s)Fn1w^(s)Fn1π)F𝑑μsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛1^𝑤𝑠superscript𝐹𝑛1𝜋𝐹differential-d𝜇\displaystyle\qquad+\int_{Y}e^{-s\varphi_{n}}v_{s}\;({\widehat{w}}(s)\circ F^{n-1}-{\widehat{w}}(s)\circ F^{n-1}\circ\pi)\circ F\,d\mu
=k=0n1Yesφnvs(Δkw^(s))Fnk1F𝑑μ+A^n(s).absentsuperscriptsubscript𝑘0𝑛1subscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠subscriptΔ𝑘^𝑤𝑠superscript𝐹𝑛𝑘1𝐹differential-d𝜇subscript^𝐴𝑛𝑠\displaystyle=\sum_{k=0}^{n-1}\int_{Y}e^{-s\varphi_{n}}v_{s}\;(\Delta_{k}{\widehat{w}}(s))\circ F^{n-k-1}\circ F\,d\mu+{\widehat{A}}_{n}(s).

Also, by Proposition 4.10(b), for each n1𝑛1n\geq 1, 0kn10𝑘𝑛10\leq k\leq n-1,

Yesφnvssubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠\displaystyle\int_{Y}e^{-s\varphi_{n}}v_{s} (Δkw^(s))Fnk1Fdμ=YesφnFnvsFn(Δkw^(s))F2nk𝑑μsubscriptΔ𝑘^𝑤𝑠superscript𝐹𝑛𝑘1𝐹𝑑𝜇subscript𝑌superscript𝑒𝑠subscript𝜑𝑛superscript𝐹𝑛subscript𝑣𝑠superscript𝐹𝑛subscriptΔ𝑘^𝑤𝑠superscript𝐹2𝑛𝑘differential-d𝜇\displaystyle\;(\Delta_{k}{\widehat{w}}(s))\circ F^{n-k-1}\circ F\,d\mu=\int_{Y}e^{-s\varphi_{n}\circ F^{n}}v_{s}\circ F^{n}\;(\Delta_{k}{\widehat{w}}(s))\circ F^{2n-k}\,d\mu
=j=0nYesφnFn(Δjvs)FnjΔkw^(s)F2nk𝑑μ+B^n,k(s)absentsuperscriptsubscript𝑗0𝑛subscript𝑌superscript𝑒𝑠subscript𝜑𝑛superscript𝐹𝑛subscriptΔ𝑗subscript𝑣𝑠superscript𝐹𝑛𝑗subscriptΔ𝑘^𝑤𝑠superscript𝐹2𝑛𝑘differential-d𝜇subscript^𝐵𝑛𝑘𝑠\displaystyle=\sum_{j=0}^{n}\int_{Y}e^{-s\varphi_{n}\circ F^{n}}(\Delta_{j}v_{s})\circ F^{n-j}\;\Delta_{k}{\widehat{w}}(s)\circ F^{2n-k}\,d\mu+{\widehat{B}}_{n,k}(s)
=j=0nY¯esφnF¯jΔjvs¯Δkw^(s)¯F¯nk+j𝑑μ¯+B^n,k(s).absentsuperscriptsubscript𝑗0𝑛subscript¯𝑌superscript𝑒𝑠subscript𝜑𝑛superscript¯𝐹𝑗¯subscriptΔ𝑗subscript𝑣𝑠¯subscriptΔ𝑘^𝑤𝑠superscript¯𝐹𝑛𝑘𝑗differential-d¯𝜇subscript^𝐵𝑛𝑘𝑠\displaystyle=\sum_{j=0}^{n}\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}e^{-s\varphi_{n}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}}\overline{\Delta_{j}v_{s}}\;\overline{\Delta_{k}{\widehat{w}}(s)}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n-k+j}\,d\bar{\mu}+{\widehat{B}}_{n,k}(s).

Next,

Y¯esφnF¯jΔjvs¯Δkw^(s)¯F¯nk+j𝑑μ¯=Y¯esφnRjΔjvs¯Δkw^(s)¯F¯nk𝑑μ¯subscript¯𝑌superscript𝑒𝑠subscript𝜑𝑛superscript¯𝐹𝑗¯subscriptΔ𝑗subscript𝑣𝑠¯subscriptΔ𝑘^𝑤𝑠superscript¯𝐹𝑛𝑘𝑗differential-d¯𝜇subscript¯𝑌superscript𝑒𝑠subscript𝜑𝑛superscript𝑅𝑗¯subscriptΔ𝑗subscript𝑣𝑠¯subscriptΔ𝑘^𝑤𝑠superscript¯𝐹𝑛𝑘differential-d¯𝜇\displaystyle\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}e^{-s\varphi_{n}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}}\overline{\Delta_{j}v_{s}}\;\overline{\Delta_{k}{\widehat{w}}(s)}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n-k+j}\,d\bar{\mu}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}e^{-s\varphi_{n}}R^{j}\overline{\Delta_{j}v_{s}}\;\overline{\Delta_{k}{\widehat{w}}(s)}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n-k}\,d\bar{\mu}
=Y¯esφnkRjΔjvs¯(esφkΔkw^(s)¯)F¯nk𝑑μ¯=Y¯R^(s)nkRjΔjvs¯W¯k(s)𝑑μ¯absentsubscript¯𝑌superscript𝑒𝑠subscript𝜑𝑛𝑘superscript𝑅𝑗¯subscriptΔ𝑗subscript𝑣𝑠superscript𝑒𝑠subscript𝜑𝑘¯subscriptΔ𝑘^𝑤𝑠superscript¯𝐹𝑛𝑘differential-d¯𝜇subscript¯𝑌^𝑅superscript𝑠𝑛𝑘superscript𝑅𝑗¯subscriptΔ𝑗subscript𝑣𝑠subscript¯𝑊𝑘𝑠differential-d¯𝜇\displaystyle\quad=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}e^{-s\varphi_{n-k}}R^{j}\overline{\Delta_{j}v_{s}}\;(e^{-s\varphi_{k}}\overline{\Delta_{k}{\widehat{w}}(s)})\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{n-k}\,d\bar{\mu}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\widehat{R}}(s)^{n-k}R^{j}\overline{\Delta_{j}v_{s}}\;{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(s)\,d\bar{\mu}
=Y¯R^(s)nk1Rj+1(esφF¯jΔjvs¯)W¯k(s)𝑑μ¯=Y¯R^(s)nk1Rj+1V¯j(s)W¯k(s)𝑑μ¯.absentsubscript¯𝑌^𝑅superscript𝑠𝑛𝑘1superscript𝑅𝑗1superscript𝑒𝑠𝜑superscript¯𝐹𝑗¯subscriptΔ𝑗subscript𝑣𝑠subscript¯𝑊𝑘𝑠differential-d¯𝜇subscript¯𝑌^𝑅superscript𝑠𝑛𝑘1superscript𝑅𝑗1subscript¯𝑉𝑗𝑠subscript¯𝑊𝑘𝑠differential-d¯𝜇\displaystyle\quad=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\widehat{R}}(s)^{n-k-1}R^{j+1}(e^{-s\varphi\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}}\overline{\Delta_{j}v_{s}})\;{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(s)\,d\bar{\mu}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\widehat{R}}(s)^{n-k-1}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(s)\;{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(s)\,d\bar{\mu}.

Altogether,

n=1Yesφnvsw^(s)Fn𝑑μ=n=1A^n(s)+n=1k=0n1B^n,k(s)+C(s)superscriptsubscript𝑛1subscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠^𝑤𝑠superscript𝐹𝑛differential-d𝜇superscriptsubscript𝑛1subscript^𝐴𝑛𝑠superscriptsubscript𝑛1superscriptsubscript𝑘0𝑛1subscript^𝐵𝑛𝑘𝑠𝐶𝑠\sum_{n=1}^{\infty}\int_{Y}e^{-s\varphi_{n}}v_{s}\,{\widehat{w}}(s)\circ F^{n}\,d\mu=\sum_{n=1}^{\infty}{\widehat{A}}_{n}(s)+\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}{\widehat{B}}_{n,k}(s)+C(s)

where

C(s)=n=1k=0n1j=0nY¯R^(s)nk1Rj+1V¯j(s)W¯k(s)𝑑μ¯.𝐶𝑠superscriptsubscript𝑛1superscriptsubscript𝑘0𝑛1superscriptsubscript𝑗0𝑛subscript¯𝑌^𝑅superscript𝑠𝑛𝑘1superscript𝑅𝑗1subscript¯𝑉𝑗𝑠subscript¯𝑊𝑘𝑠differential-d¯𝜇C(s)=\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\sum_{j=0}^{n}\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\widehat{R}}(s)^{n-k-1}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(s)\;{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(s)\,d\bar{\mu}.

Now

n=1k=0n1j=0nR^(s)nk1ajbk=0jkn=k+1R^(s)nk1ajbk+j>k0n=jR^(s)nk1ajbksuperscriptsubscript𝑛1superscriptsubscript𝑘0𝑛1superscriptsubscript𝑗0𝑛^𝑅superscript𝑠𝑛𝑘1subscript𝑎𝑗subscript𝑏𝑘subscript0𝑗𝑘superscriptsubscript𝑛𝑘1^𝑅superscript𝑠𝑛𝑘1subscript𝑎𝑗subscript𝑏𝑘subscript𝑗𝑘0superscriptsubscript𝑛𝑗^𝑅superscript𝑠𝑛𝑘1subscript𝑎𝑗subscript𝑏𝑘\displaystyle\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\sum_{j=0}^{n}{\widehat{R}}(s)^{n-k-1}a_{j}\;b_{k}=\sum_{0\leq j\leq k}\sum_{n=k+1}^{\infty}{\widehat{R}}(s)^{n-k-1}a_{j}\;b_{k}+\sum_{j>k\geq 0}\sum_{n=j}^{\infty}{\widehat{R}}(s)^{n-k-1}a_{j}\;b_{k}
=k=0j=0kT^(s)ajbk+j=1k=0j1R^(s)jk1T^(s)ajbk=j,k=0R^(s)max{jk1,0}T^(s)ajbk.absentsuperscriptsubscript𝑘0superscriptsubscript𝑗0𝑘^𝑇𝑠subscript𝑎𝑗subscript𝑏𝑘superscriptsubscript𝑗1superscriptsubscript𝑘0𝑗1^𝑅superscript𝑠𝑗𝑘1^𝑇𝑠subscript𝑎𝑗subscript𝑏𝑘superscriptsubscript𝑗𝑘0^𝑅superscript𝑠𝑗𝑘10^𝑇𝑠subscript𝑎𝑗subscript𝑏𝑘\displaystyle=\sum_{k=0}^{\infty}\sum_{j=0}^{k}{\widehat{T}}(s)a_{j}\;b_{k}+\sum_{j=1}^{\infty}\sum_{k=0}^{j-1}{\widehat{R}}(s)^{j-k-1}{\widehat{T}}(s)a_{j}\;b_{k}=\sum_{j,k=0}^{\infty}{\widehat{R}}(s)^{\max\{j-k-1,0\}}{\widehat{T}}(s)a_{j}\;b_{k}.

This completes the proof. ∎

For wL(Yφ)𝑤superscript𝐿superscript𝑌𝜑w\in L^{\infty}(Y^{\varphi}), we define the approximation operators

Δ~kw(y,u)subscript~Δ𝑘𝑤𝑦𝑢\displaystyle\widetilde{\Delta}_{k}w(y,u) ={w(Fkπy,u)w(Fk1πFy,u)k1w(πy,u)k=0,absentcases𝑤superscript𝐹𝑘𝜋𝑦𝑢𝑤superscript𝐹𝑘1𝜋𝐹𝑦𝑢𝑘1𝑤𝜋𝑦𝑢𝑘0\displaystyle=\begin{cases}w(F^{k}\pi y,u)-w(F^{k-1}\pi Fy,u)&k\geq 1\\ w(\pi y,u)&k=0\end{cases},
E~kw(y,u)subscript~𝐸𝑘𝑤𝑦𝑢\displaystyle\widetilde{E}_{k}w(y,u) =w(Fky,u)w(Fkπy,u),k0,formulae-sequenceabsent𝑤superscript𝐹𝑘𝑦𝑢𝑤superscript𝐹𝑘𝜋𝑦𝑢𝑘0\displaystyle=w(F^{k}y,u)-w(F^{k}\pi y,u),\,k\geq 0,

for yY𝑦𝑌y\in Y, u[0,φ(Fky)]𝑢0𝜑superscript𝐹𝑘𝑦u\in[0,\varphi(F^{k}y)].

Proposition 4.12

(a) Let wγ(Yφ)𝑤subscript𝛾superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}), k0𝑘0k\geq 0. Then for all yY𝑦𝑌y\in Y, u[0,φ(Fky)]𝑢0𝜑superscript𝐹𝑘𝑦u\in[0,\varphi(F^{k}y)],

|Δ~kw(y,u)|2C2γ1k1wγφ(Fky)ηand|E~kw(y,u)|2C2γ1k|w|γφ(Fky)η.formulae-sequencesubscript~Δ𝑘𝑤𝑦𝑢2subscript𝐶2superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾𝜑superscriptsuperscript𝐹𝑘𝑦𝜂andsubscript~𝐸𝑘𝑤𝑦𝑢2subscript𝐶2superscriptsubscript𝛾1𝑘subscript𝑤𝛾𝜑superscriptsuperscript𝐹𝑘𝑦𝜂|\widetilde{\Delta}_{k}w(y,u)|\leq 2C_{2}\gamma_{1}^{k-1}\|w\|_{\gamma}\varphi(F^{k}y)^{\eta}\quad\text{and}\quad|\widetilde{E}_{k}w(y,u)|\leq 2C_{2}\gamma_{1}^{k}|w|_{\gamma}\varphi(F^{k}y)^{\eta}.

(b) Let wγ(Yφ)𝑤subscript𝛾superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}), k0𝑘0k\geq 0. Then for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, u[0,φ(Fky)][0,φ(Fky)]𝑢0𝜑superscript𝐹𝑘𝑦0𝜑superscript𝐹𝑘superscript𝑦u\in[0,\varphi(F^{k}y)]\cap[0,\varphi(F^{k}y^{\prime})],

|Δ~kw(y,u)Δ~kw(y,u)|4C2γ1s(y,y)k|w|γφ(Fky)η.subscript~Δ𝑘𝑤𝑦𝑢subscript~Δ𝑘𝑤superscript𝑦𝑢4subscript𝐶2superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑘subscript𝑤𝛾𝜑superscriptsuperscript𝐹𝑘𝑦𝜂|\widetilde{\Delta}_{k}w(y,u)-\widetilde{\Delta}_{k}w(y^{\prime},u)|\leq 4C_{2}\gamma_{1}^{s(y,y^{\prime})-k}|w|_{\gamma}\varphi(F^{k}y)^{\eta}.

(c) Let wγ,η(Yφ)𝑤subscript𝛾𝜂superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}), k0𝑘0k\geq 0. Then for all yY𝑦𝑌y\in Y, u,u[0,φ(Fky)]𝑢superscript𝑢0𝜑superscript𝐹𝑘𝑦u,u^{\prime}\in[0,\varphi(F^{k}y)],

|Δ~kw(y,u)Δ~kw(y,u)|2|w|,η|uu|η.subscript~Δ𝑘𝑤𝑦𝑢subscript~Δ𝑘𝑤𝑦superscript𝑢2subscript𝑤𝜂superscript𝑢superscript𝑢𝜂|\widetilde{\Delta}_{k}w(y,u)-\widetilde{\Delta}_{k}w(y,u^{\prime})|\leq 2|w|_{\infty,\eta}|u-u^{\prime}|^{\eta}.
Proof.

(a) Clearly |Δ~0w(y,u)||w|subscript~Δ0𝑤𝑦𝑢subscript𝑤|\widetilde{\Delta}_{0}w(y,u)|\leq|w|_{\infty}. By (3.1), for k1𝑘1k\geq 1,

|Δ~kw(y,u)|subscript~Δ𝑘𝑤𝑦𝑢\displaystyle|\widetilde{\Delta}_{k}w(y,u)| |w|γφ(Fky)(d(Fkπy,Fk1πFy)+γs(Fkπy,Fk1πFy))absentsubscript𝑤𝛾𝜑superscript𝐹𝑘𝑦𝑑superscript𝐹𝑘𝜋𝑦superscript𝐹𝑘1𝜋𝐹𝑦superscript𝛾𝑠superscript𝐹𝑘𝜋𝑦superscript𝐹𝑘1𝜋𝐹𝑦\displaystyle\leq|w|_{\gamma}\varphi(F^{k}y)(d(F^{k}\pi y,F^{k-1}\pi Fy)+\gamma^{s(F^{k}\pi y,F^{k-1}\pi Fy)})
=|w|γφ(Fky)d(Fkπy,Fk1πFy)C2γk1|w|γφ(Fky).absentsubscript𝑤𝛾𝜑superscript𝐹𝑘𝑦𝑑superscript𝐹𝑘𝜋𝑦superscript𝐹𝑘1𝜋𝐹𝑦subscript𝐶2superscript𝛾𝑘1subscript𝑤𝛾𝜑superscript𝐹𝑘𝑦\displaystyle=|w|_{\gamma}\varphi(F^{k}y)d(F^{k}\pi y,F^{k-1}\pi Fy)\leq C_{2}\gamma^{k-1}|w|_{\gamma}\varphi(F^{k}y).

Also, |Δ~kw|2|w|subscript~Δ𝑘𝑤2subscript𝑤|\widetilde{\Delta}_{k}w|\leq 2|w|_{\infty}, so

|Δ~kw(y,u)|2C2wγmin{1,γk1φ(Fky)}2C2γ1k1wγφ(Fky)η.subscript~Δ𝑘𝑤𝑦𝑢2subscript𝐶2subscriptnorm𝑤𝛾1superscript𝛾𝑘1𝜑superscript𝐹𝑘𝑦2subscript𝐶2superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾𝜑superscriptsuperscript𝐹𝑘𝑦𝜂|\widetilde{\Delta}_{k}w(y,u)|\leq 2C_{2}\|w\|_{\gamma}\min\{1,\gamma^{k-1}\varphi(F^{k}y)\}\leq 2C_{2}\gamma_{1}^{k-1}\|w\|_{\gamma}\varphi(F^{k}y)^{\eta}.

This proves the estimate for Δ~kwsubscript~Δ𝑘𝑤\widetilde{\Delta}_{k}w, and the estimate for E~kwsubscript~𝐸𝑘𝑤\widetilde{E}_{k}w is similar.
(b) First suppose that k1𝑘1k\geq 1 and note by (3.2) that

d(Fkπy,Fkπy)C2γs(y,y)k,d(Fk1πFy,Fk1πFy)C2γs(y,y)k.formulae-sequence𝑑superscript𝐹𝑘𝜋𝑦superscript𝐹𝑘𝜋superscript𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘𝑑superscript𝐹𝑘1𝜋𝐹𝑦superscript𝐹𝑘1𝜋𝐹superscript𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘\displaystyle d(F^{k}\pi y,F^{k}\pi y^{\prime})\leq C_{2}\gamma^{s(y,y^{\prime})-k},\quad d(F^{k-1}\pi Fy,F^{k-1}\pi Fy^{\prime})\leq C_{2}\gamma^{s(y,y^{\prime})-k}.

It follows that

|w(Fkπy,u)\displaystyle|w(F^{k}\pi y,u) w(Fkπy,u)||w|γφ(Fky)(d(Fkπy,Fkπy)+γs(Fkπy,Fkπy))\displaystyle-w(F^{k}\pi y^{\prime},u)|\leq|w|_{\gamma}\varphi(F^{k}y)(d(F^{k}\pi y,F^{k}\pi y^{\prime})+\gamma^{s(F^{k}\pi y,F^{k}\pi y^{\prime})})
|w|γφ(Fky)(C2γs(y,y)k+γs(y,y)k)2C2γs(y,y)k|w|γφ(Fky).absentsubscript𝑤𝛾𝜑superscript𝐹𝑘𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘superscript𝛾𝑠𝑦superscript𝑦𝑘2subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘subscript𝑤𝛾𝜑superscript𝐹𝑘𝑦\displaystyle\leq|w|_{\gamma}\varphi(F^{k}y)(C_{2}\gamma^{s(y,y^{\prime})-k}+\gamma^{s(y,y^{\prime})-k})\leq 2C_{2}\gamma^{s(y,y^{\prime})-k}|w|_{\gamma}\varphi(F^{k}y).

Similarly, |w(Fk1πFy,u)w(Fk1πFy,u)|2C2γs(y,y)k|w|γφ(Fky)𝑤superscript𝐹𝑘1𝜋𝐹𝑦𝑢𝑤superscript𝐹𝑘1𝜋𝐹superscript𝑦𝑢2subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘subscript𝑤𝛾𝜑superscript𝐹𝑘𝑦|w(F^{k-1}\pi Fy,u)-w(F^{k-1}\pi Fy^{\prime},u)|\leq 2C_{2}\gamma^{s(y,y^{\prime})-k}|w|_{\gamma}\varphi(F^{k}y). Hence

|Δ~kw(y,u)Δ~kw(y,u)|subscript~Δ𝑘𝑤𝑦𝑢subscript~Δ𝑘𝑤superscript𝑦𝑢\displaystyle|\widetilde{\Delta}_{k}w(y,u)-\widetilde{\Delta}_{k}w(y^{\prime},u)| |w(Fkπy,u)w(Fkπy,u)|absent𝑤superscript𝐹𝑘𝜋𝑦𝑢𝑤superscript𝐹𝑘𝜋superscript𝑦𝑢\displaystyle\leq|w(F^{k}\pi y,u)-w(F^{k}\pi y^{\prime},u)|
+|w(Fk1πFy,u)w(Fk1πFy,u)|𝑤superscript𝐹𝑘1𝜋𝐹𝑦𝑢𝑤superscript𝐹𝑘1𝜋𝐹superscript𝑦𝑢\displaystyle\qquad+|w(F^{k-1}\pi Fy,u)-w(F^{k-1}\pi Fy^{\prime},u)|
4C2γs(y,y)k|w|γφ(Fky).absent4subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦𝑘subscript𝑤𝛾𝜑superscript𝐹𝑘𝑦\displaystyle\leq 4C_{2}\gamma^{s(y,y^{\prime})-k}|w|_{\gamma}\varphi(F^{k}y).

Also, |Δ~kw(y,u)Δ~kw(y,u)|4|w|subscript~Δ𝑘𝑤𝑦𝑢subscript~Δ𝑘𝑤superscript𝑦𝑢4subscript𝑤|\widetilde{\Delta}_{k}w(y,u)-\widetilde{\Delta}_{k}w(y^{\prime},u)|\leq 4|w|_{\infty}, so

|Δ~kw(y,u)Δ~kw(y,u)|4C2γ1s(y,y)k|w|γφ(Fky)η.subscript~Δ𝑘𝑤𝑦𝑢subscript~Δ𝑘𝑤superscript𝑦𝑢4subscript𝐶2superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑘subscript𝑤𝛾𝜑superscriptsuperscript𝐹𝑘𝑦𝜂|\widetilde{\Delta}_{k}w(y,u)-\widetilde{\Delta}_{k}w(y^{\prime},u)|\leq 4C_{2}\gamma_{1}^{s(y,y^{\prime})-k}|w|_{\gamma}\varphi(F^{k}y)^{\eta}.

The case k=0𝑘0k=0 is the same with one term omitted.
(c) For k1𝑘1k\geq 1,

|Δ~kw(y,u)\displaystyle|\widetilde{\Delta}_{k}w(y,u)- Δ~kw(y,u)||w(Fkπy,u)w(Fkπy,u)|\displaystyle\widetilde{\Delta}_{k}w(y,u^{\prime})|\leq|w(F^{k}\pi y,u)-w(F^{k}\pi y,u^{\prime})|
+|w(Fk1πFy,u)w(Fk1πFy,u)|2|w|,η|uu|η.𝑤superscript𝐹𝑘1𝜋𝐹𝑦𝑢𝑤superscript𝐹𝑘1𝜋𝐹𝑦superscript𝑢2subscript𝑤𝜂superscript𝑢superscript𝑢𝜂\displaystyle+|w(F^{k-1}\pi Fy,u)-w(F^{k-1}\pi Fy,u^{\prime})|\leq 2|w|_{\infty,\eta}|u-u^{\prime}|^{\eta}.

The case k=0𝑘0k=0 is the same with one term omitted. ∎

We end this subsection by noting for all k0𝑘0k\geq 0 the identities

Δkvs(y)subscriptΔ𝑘subscript𝑣𝑠𝑦\displaystyle\Delta_{k}v_{s}(y) =0φ(Fky)esuΔ~kv(y,u)𝑑u,absentsuperscriptsubscript0𝜑superscript𝐹𝑘𝑦superscript𝑒𝑠𝑢subscript~Δ𝑘𝑣𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{k}y)}e^{su}\widetilde{\Delta}_{k}v(y,u)\,du, Δkw^(s)(y)subscriptΔ𝑘^𝑤𝑠𝑦\displaystyle\qquad\Delta_{k}{\widehat{w}}(s)(y) =0φ(Fky)esuΔ~kw(y,u)𝑑u,absentsuperscriptsubscript0𝜑superscript𝐹𝑘𝑦superscript𝑒𝑠𝑢subscript~Δ𝑘𝑤𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{k}y)}e^{-su}\widetilde{\Delta}_{k}w(y,u)\,du,
Ekvs(y)subscript𝐸𝑘subscript𝑣𝑠𝑦\displaystyle E_{k}v_{s}(y) =0φ(Fky)esuE~kv(y,u)𝑑u,absentsuperscriptsubscript0𝜑superscript𝐹𝑘𝑦superscript𝑒𝑠𝑢subscript~𝐸𝑘𝑣𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{k}y)}e^{su}\widetilde{E}_{k}v(y,u)\,du, Ekw^(s)(y)subscript𝐸𝑘^𝑤𝑠𝑦\displaystyle\qquad E_{k}{\widehat{w}}(s)(y) =0φ(Fky)esuE~kw(y,u)𝑑u.absentsuperscriptsubscript0𝜑superscript𝐹𝑘𝑦superscript𝑒𝑠𝑢subscript~𝐸𝑘𝑤𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{k}y)}e^{-su}\widetilde{E}_{k}w(y,u)\,du.

4.3 Estimates for Ansubscript𝐴𝑛A_{n} and Bn,ksubscript𝐵𝑛𝑘B_{n,k}

We continue to suppose that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) where β>1𝛽1\beta>1, and that q𝑞q, η𝜂\eta, γ1subscript𝛾1\gamma_{1}, θ𝜃\theta are as in Subsection 4.1. Let c=1/(2C1)superscript𝑐12subscript𝐶1c^{\prime}=1/(2C_{1}). As shown in the proofs of Propositions 4.14 and 4.15 below, A^nsubscript^𝐴𝑛{\widehat{A}}_{n} and B^n,ksubscript^𝐵𝑛𝑘{\widehat{B}}_{n,k} are Laplace transforms of Lsuperscript𝐿L^{\infty} functions An,Bn,k:[0,):subscript𝐴𝑛subscript𝐵𝑛𝑘0A_{n},\,B_{n,k}:[0,\infty)\to{\mathbb{R}}. In this subsection, we obtain estimates for these functions An,Bn,ksubscript𝐴𝑛subscript𝐵𝑛𝑘A_{n},\,B_{n,k}.

Proposition 4.13

There is a constant C>0𝐶0C>0 such that

YφφFn1{φn+1>t}𝑑μCnYφ1{φ>ct/n}𝑑μfor all n1t>0.subscript𝑌𝜑𝜑superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡differential-d𝜇𝐶𝑛subscript𝑌𝜑subscript1𝜑superscript𝑐𝑡𝑛differential-d𝜇for all n1t>0.\textstyle\int_{Y}\varphi\,\varphi\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu\leq Cn\int_{Y}\varphi 1_{\{\varphi>c^{\prime}t/n\}}\,d\mu\quad\text{for all $n\geq 1$, $t>0$.}
Proof.

Since F𝐹F is Gibbs-Markov, there is a constant C0subscript𝐶0C_{0} (called C2subscript𝐶2C_{2} in [26]) such that

|R(φ1{φ>c})|subscript𝑅𝜑subscript1𝜑𝑐\displaystyle|R(\varphi 1_{\{\varphi>c\}})|_{\infty} C0μ(Yj)|1Yjφ|1{|1Yjφ|>c}absentsubscript𝐶0𝜇subscript𝑌𝑗subscriptsubscript1subscript𝑌𝑗𝜑subscript1subscriptsubscript1subscript𝑌𝑗𝜑𝑐\displaystyle\leq C_{0}\sum\mu(Y_{j})|1_{Y_{j}}\varphi|_{\infty}1_{\{|1_{Y_{j}}\varphi|_{\infty}>c\}}
2C0C1μ(Yj)infYjφ1{infYjφ>cc}KYφ1{φ>cc}𝑑μ,absent2subscript𝐶0subscript𝐶1𝜇subscript𝑌𝑗subscriptinfimumsubscript𝑌𝑗𝜑subscript1subscriptinfimumsubscript𝑌𝑗𝜑superscript𝑐𝑐𝐾subscript𝑌𝜑subscript1𝜑superscript𝑐𝑐differential-d𝜇\displaystyle\leq 2C_{0}C_{1}\sum\mu(Y_{j}){\textstyle\inf_{Y_{j}}}\varphi 1_{\{{\textstyle\inf_{Y_{j}}}\varphi>c^{\prime}c\}}\leq K{\textstyle\int}_{Y}\varphi 1_{\{\varphi>c^{\prime}c\}}\,d\mu,

where K=2C0C1𝐾2subscript𝐶0subscript𝐶1K=2C_{0}C_{1}. Similarly, |Rφ|K|φ|1subscript𝑅𝜑𝐾subscript𝜑1|R\varphi|_{\infty}\leq K|\varphi|_{1} and |R1{φ>c}|Kμ(φ>cc)subscript𝑅subscript1𝜑𝑐𝐾𝜇𝜑superscript𝑐𝑐|R1_{\{\varphi>c\}}|_{\infty}\leq K\mu(\varphi>c^{\prime}c).

Now

Yφsubscript𝑌𝜑\displaystyle\int_{Y}\varphi φFn1{φn+1>t}dμj=0nYφFnφ1{φFj>t/n}𝑑μ𝜑superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡𝑑𝜇superscriptsubscript𝑗0𝑛subscript𝑌𝜑superscript𝐹𝑛𝜑subscript1𝜑superscript𝐹𝑗𝑡𝑛differential-d𝜇\displaystyle\varphi\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu\leq\sum_{j=0}^{n}\int_{Y}\varphi\circ F^{n}\,\varphi 1_{\{\varphi\circ F^{j}>t/n\}}\,d\mu
=j=0nYφRn(φ1{φFj>t/n})𝑑μ=j=0nYφRnj(1{φ>t/n}Rjφ)𝑑μ.absentsuperscriptsubscript𝑗0𝑛subscript𝑌𝜑superscript𝑅𝑛𝜑subscript1𝜑superscript𝐹𝑗𝑡𝑛differential-d𝜇superscriptsubscript𝑗0𝑛subscript𝑌𝜑superscript𝑅𝑛𝑗subscript1𝜑𝑡𝑛superscript𝑅𝑗𝜑differential-d𝜇\displaystyle=\sum_{j=0}^{n}\int_{Y}\varphi\,R^{n}(\varphi 1_{\{\varphi\circ F^{j}>t/n\}})\,d\mu=\sum_{j=0}^{n}\int_{Y}\varphi\,R^{n-j}(1_{\{\varphi>t/n\}}R^{j}\varphi)\,d\mu.

For 1jn11𝑗𝑛11\leq j\leq n-1,

|Yφ\displaystyle\textstyle\Big{|}\int_{Y}\varphi\, Rnj(1{φ>t/n}Rjφ)dμ||φ|1|Rnj(1{φ>t/n}Rjφ)|\displaystyle R^{n-j}(1_{\{\varphi>t/n\}}R^{j}\varphi)\,d\mu\Big{|}\leq|\varphi|_{1}|R^{n-j}(1_{\{\varphi>t/n\}}R^{j}\varphi)|_{\infty}
|φ|1|Rjφ||Rnj1{φ>t/n}||φ|1|Rφ||R1{φ>t/n}|K2|φ|12μ(φ>ct/n).absentsubscript𝜑1subscriptsuperscript𝑅𝑗𝜑subscriptsuperscript𝑅𝑛𝑗subscript1𝜑𝑡𝑛subscript𝜑1subscript𝑅𝜑subscript𝑅subscript1𝜑𝑡𝑛superscript𝐾2superscriptsubscript𝜑12𝜇𝜑superscript𝑐𝑡𝑛\displaystyle\textstyle\leq|\varphi|_{1}|R^{j}\varphi|_{\infty}|R^{n-j}1_{\{\varphi>t/n\}}|_{\infty}\leq|\varphi|_{1}|R\varphi|_{\infty}|R1_{\{\varphi>t/n\}}|_{\infty}\leq K^{2}|\varphi|_{1}^{2}\mu(\varphi>c^{\prime}t/n).

For j=n𝑗𝑛j=n,

|YφRnj(1{φ>t/n}Rjφ)𝑑μ||Rφ|Yφ 1{φ>t/n}𝑑μK|φ|1Yφ1{φ>ct/n}𝑑μ.subscript𝑌𝜑superscript𝑅𝑛𝑗subscript1𝜑𝑡𝑛superscript𝑅𝑗𝜑differential-d𝜇subscript𝑅𝜑subscript𝑌𝜑subscript1𝜑𝑡𝑛differential-d𝜇𝐾subscript𝜑1subscript𝑌𝜑subscript1𝜑superscript𝑐𝑡𝑛differential-d𝜇\textstyle|\int_{Y}\varphi\,R^{n-j}(1_{\{\varphi>t/n\}}R^{j}\varphi)\,d\mu|\leq|R\varphi|_{\infty}\int_{Y}\varphi\,1_{\{\varphi>t/n\}}\,d\mu\leq K|\varphi|_{1}\int_{Y}\varphi 1_{\{\varphi>c^{\prime}t/n\}}\,d\mu.

Finally for j=0𝑗0j=0,

|YφRnj(1{φ>t/n}Rjφ)𝑑μ||φ|1|R(φ 1{φ>t/n})|K|φ|1Yφ1{φ>ct/n}𝑑μ,subscript𝑌𝜑superscript𝑅𝑛𝑗subscript1𝜑𝑡𝑛superscript𝑅𝑗𝜑differential-d𝜇subscript𝜑1subscript𝑅𝜑subscript1𝜑𝑡𝑛𝐾subscript𝜑1subscript𝑌𝜑subscript1𝜑superscript𝑐𝑡𝑛differential-d𝜇\textstyle|\int_{Y}\varphi\,R^{n-j}(1_{\{\varphi>t/n\}}R^{j}\varphi)\,d\mu|\leq|\varphi|_{1}|R(\varphi\,1_{\{\varphi>t/n\}})|_{\infty}\leq K|\varphi|_{1}\int_{Y}\varphi 1_{\{\varphi>c^{\prime}t/n\}}\,d\mu,

completing the proof. ∎

Proposition 4.14

There is a constant C>0𝐶0C>0 such that

|An(t)|Cnβγ1n|v||w|γ(t+1)(β1)for all vL(Yφ)wγ(Yφ)n1t>0.subscript𝐴𝑛𝑡𝐶superscript𝑛𝛽superscriptsubscript𝛾1𝑛subscript𝑣subscript𝑤𝛾superscript𝑡1𝛽1for all vL(Yφ)wγ(Yφ)n1t>0.|A_{n}(t)|\leq Cn^{\beta}\gamma_{1}^{n}|v|_{\infty}|w|_{\gamma}\,(t+1)^{-(\beta-1)}\;\text{for all $v\in L^{\infty}(Y^{\varphi})$, $w\in{\mathcal{H}}_{\gamma}(Y^{\varphi})$, $n\geq 1$, $t>0$.}
Proof.

We compute that

A^n(s)subscript^𝐴𝑛𝑠\displaystyle{\widehat{A}}_{n}(s) =Yesφnvs(En1w^(s))F𝑑μabsentsubscript𝑌superscript𝑒𝑠subscript𝜑𝑛subscript𝑣𝑠subscript𝐸𝑛1^𝑤𝑠𝐹differential-d𝜇\displaystyle=\int_{Y}e^{-s\varphi_{n}}v_{s}\;(E_{n-1}{\widehat{w}}(s))\circ F\,d\mu
=Y0φ(y)v(y,u)0φ(Fny)es(φn(y)u+u)E~n1w(Fy,u)𝑑u𝑑u𝑑μabsentsubscript𝑌superscriptsubscript0𝜑𝑦𝑣𝑦𝑢superscriptsubscript0𝜑superscript𝐹𝑛𝑦superscript𝑒𝑠subscript𝜑𝑛𝑦𝑢superscript𝑢subscript~𝐸𝑛1𝑤𝐹𝑦superscript𝑢differential-dsuperscript𝑢differential-d𝑢differential-d𝜇\displaystyle=\int_{Y}\int_{0}^{\varphi(y)}v(y,u)\int_{0}^{\varphi(F^{n}y)}e^{-s(\varphi_{n}(y)-u+u^{\prime})}\widetilde{E}_{n-1}w(Fy,u^{\prime})\,du^{\prime}\,du\,d\mu
=Y0φ(y)v(y,u)φn(y)uφn+1(y)uestE~n1w(Fy,tφn(y)+u)𝑑t𝑑u𝑑μ.absentsubscript𝑌superscriptsubscript0𝜑𝑦𝑣𝑦𝑢superscriptsubscriptsubscript𝜑𝑛𝑦𝑢subscript𝜑𝑛1𝑦𝑢superscript𝑒𝑠𝑡subscript~𝐸𝑛1𝑤𝐹𝑦𝑡subscript𝜑𝑛𝑦𝑢differential-d𝑡differential-d𝑢differential-d𝜇\displaystyle=\int_{Y}\int_{0}^{\varphi(y)}v(y,u)\int_{\varphi_{n}(y)-u}^{\varphi_{n+1}(y)-u}e^{-st}\widetilde{E}_{n-1}w(Fy,t-\varphi_{n}(y)+u)\,dt\,du\,d\mu.

Hence

An(t)=Y0φ(y)v(y,u)1{φn(y)u<t<φn+1(y)u}E~n1w(Fy,tφn(y)+u)𝑑u𝑑μ.subscript𝐴𝑛𝑡subscript𝑌superscriptsubscript0𝜑𝑦𝑣𝑦𝑢subscript1subscript𝜑𝑛𝑦𝑢𝑡subscript𝜑𝑛1𝑦𝑢subscript~𝐸𝑛1𝑤𝐹𝑦𝑡subscript𝜑𝑛𝑦𝑢differential-d𝑢differential-d𝜇\displaystyle A_{n}(t)=\int_{Y}\int_{0}^{\varphi(y)}v(y,u)1_{\{\varphi_{n}(y)-u<t<\varphi_{n+1}(y)-u\}}\widetilde{E}_{n-1}w(Fy,t-\varphi_{n}(y)+u)\,du\,d\mu.

By Proposition 4.12(a), |E~n1w(Fy,tφn(y)+u)|2C2γ1n1|w|γφ(Fny)ηsubscript~𝐸𝑛1𝑤𝐹𝑦𝑡subscript𝜑𝑛𝑦𝑢2subscript𝐶2superscriptsubscript𝛾1𝑛1subscript𝑤𝛾𝜑superscriptsuperscript𝐹𝑛𝑦𝜂|\widetilde{E}_{n-1}w(Fy,t-\varphi_{n}(y)+u)|\leq 2C_{2}\gamma_{1}^{n-1}|w|_{\gamma}\varphi(F^{n}y)^{\eta} and so

|An(t)|2C2γ1n1|v||w|γYφφFn1{φn+1>t}𝑑μ.subscript𝐴𝑛𝑡2subscript𝐶2superscriptsubscript𝛾1𝑛1subscript𝑣subscript𝑤𝛾subscript𝑌𝜑𝜑superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡differential-d𝜇\textstyle|A_{n}(t)|\leq 2C_{2}\gamma_{1}^{n-1}|v|_{\infty}|w|_{\gamma}\int_{Y}\varphi\,\varphi\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu.

The result follows from Propositions 3.4(b) (with η=1𝜂1\eta=1) and 4.13. ∎

Proposition 4.15

There is a constant C>0𝐶0C>0 such that

|Bn,k(t)|Cnβγ1n|v|γ|w|(t+1)(β1)for all vγ(Yφ)wL(Yφ)n1k0t>0.subscript𝐵𝑛𝑘𝑡𝐶superscript𝑛𝛽superscriptsubscript𝛾1𝑛subscript𝑣𝛾subscript𝑤superscript𝑡1𝛽1for all vγ(Yφ)wL(Yφ)n1k0t>0.|B_{n,k}(t)|\leq Cn^{\beta}\gamma_{1}^{n}|v|_{\gamma}|w|_{\infty}\,(t+1)^{-(\beta-1)}\;\text{for all $v\in{\mathcal{H}}_{\gamma}(Y^{\varphi})$, $w\in L^{\infty}(Y^{\varphi})$, $n\geq 1$, $k\geq 0$, $t>0$.}
Proof.

We compute that

B^n,k(s)=YesφnFnEnvs(Δkw^(s))F2nk𝑑μsubscript^𝐵𝑛𝑘𝑠subscript𝑌superscript𝑒𝑠subscript𝜑𝑛superscript𝐹𝑛subscript𝐸𝑛subscript𝑣𝑠subscriptΔ𝑘^𝑤𝑠superscript𝐹2𝑛𝑘differential-d𝜇\displaystyle{\widehat{B}}_{n,k}(s)=\int_{Y}e^{-s\varphi_{n}\circ F^{n}}E_{n}v_{s}\;(\Delta_{k}{\widehat{w}}(s))\circ F^{2n-k}\,d\mu
=Y0φ(F2ny)0φ(Fny)es(φn(Fny)u+u)E~nv(y,u)Δ~kw(F2nky,u)𝑑u𝑑u𝑑μabsentsubscript𝑌superscriptsubscript0𝜑superscript𝐹2𝑛𝑦superscriptsubscript0𝜑superscript𝐹𝑛𝑦superscript𝑒𝑠subscript𝜑𝑛superscript𝐹𝑛𝑦superscript𝑢𝑢subscript~𝐸𝑛𝑣𝑦superscript𝑢subscript~Δ𝑘𝑤superscript𝐹2𝑛𝑘𝑦𝑢differential-dsuperscript𝑢differential-d𝑢differential-d𝜇\displaystyle\quad=\int_{Y}\int_{0}^{\varphi(F^{2n}y)}\int_{0}^{\varphi(F^{n}y)}e^{-s(\varphi_{n}(F^{n}y)-u^{\prime}+u)}\widetilde{E}_{n}v(y,u^{\prime})\widetilde{\Delta}_{k}w(F^{2n-k}y,u)\,du^{\prime}\,du\,d\mu
=Y0φ(F2ny)φn1(Fn+1y)+uφn(Fny)+uestE~nv(y,φn(Fny)t+u)Δ~kw(F2nky,u)𝑑t𝑑u𝑑μ.absentsubscript𝑌superscriptsubscript0𝜑superscript𝐹2𝑛𝑦superscriptsubscriptsubscript𝜑𝑛1superscript𝐹𝑛1𝑦𝑢subscript𝜑𝑛superscript𝐹𝑛𝑦𝑢superscript𝑒𝑠𝑡subscript~𝐸𝑛𝑣𝑦subscript𝜑𝑛superscript𝐹𝑛𝑦𝑡𝑢subscript~Δ𝑘𝑤superscript𝐹2𝑛𝑘𝑦𝑢differential-d𝑡differential-d𝑢differential-d𝜇\displaystyle\quad=\int_{Y}\int_{0}^{\varphi(F^{2n}y)}\int_{\varphi_{n-1}(F^{n+1}y)+u}^{\varphi_{n}(F^{n}y)+u}e^{-st}\widetilde{E}_{n}v(y,\varphi_{n}(F^{n}y)-t+u)\widetilde{\Delta}_{k}w(F^{2n-k}y,u)\,dt\,du\,d\mu.

Hence

Bn,k(t)subscript𝐵𝑛𝑘𝑡\displaystyle B_{n,k}(t) =Y0φ(F2ny)1{φn1(Fn+1y)+u<t<φn(Fny)+u}absentsubscript𝑌superscriptsubscript0𝜑superscript𝐹2𝑛𝑦subscript1subscript𝜑𝑛1superscript𝐹𝑛1𝑦𝑢𝑡subscript𝜑𝑛superscript𝐹𝑛𝑦𝑢\displaystyle=\int_{Y}\int_{0}^{\varphi(F^{2n}y)}1_{\{\varphi_{n-1}(F^{n+1}y)+u<t<\varphi_{n}(F^{n}y)+u\}}
×E~nv(y,φn(Fny)t+u)Δ~kw(F2nky,u)dudμ.absentsubscript~𝐸𝑛𝑣𝑦subscript𝜑𝑛superscript𝐹𝑛𝑦𝑡𝑢subscript~Δ𝑘𝑤superscript𝐹2𝑛𝑘𝑦𝑢𝑑𝑢𝑑𝜇\displaystyle\qquad\qquad\qquad\qquad\times\widetilde{E}_{n}v(y,\varphi_{n}(F^{n}y)-t+u)\widetilde{\Delta}_{k}w(F^{2n-k}y,u)\,du\,d\mu.

By Proposition 4.12(a), |E~nv(y,φn(Fny)t+u)|2C2γ1n|v|γφ(Fny)subscript~𝐸𝑛𝑣𝑦subscript𝜑𝑛superscript𝐹𝑛𝑦𝑡𝑢2subscript𝐶2superscriptsubscript𝛾1𝑛subscript𝑣𝛾𝜑superscript𝐹𝑛𝑦|\widetilde{E}_{n}v(y,\varphi_{n}(F^{n}y)-t+u)|\leq 2C_{2}\gamma_{1}^{n}|v|_{\gamma}\varphi(F^{n}y). Also |Δ~kw(F2nky,u)|2|w|subscript~Δ𝑘𝑤superscript𝐹2𝑛𝑘𝑦𝑢2subscript𝑤|\widetilde{\Delta}_{k}w(F^{2n-k}y,u)|\leq 2|w|_{\infty}. Hence

|Bn,k(t)|subscript𝐵𝑛𝑘𝑡\displaystyle|B_{n,k}(t)| 2C2γ1n|v|γ|w|YφF2nφFn1{φn+1Fn>t}𝑑μabsent2subscript𝐶2superscriptsubscript𝛾1𝑛subscript𝑣𝛾subscript𝑤subscript𝑌𝜑superscript𝐹2𝑛𝜑superscript𝐹𝑛subscript1subscript𝜑𝑛1superscript𝐹𝑛𝑡differential-d𝜇\displaystyle\leq 2C_{2}\gamma_{1}^{n}|v|_{\gamma}|w|_{\infty}\int_{Y}\varphi\circ F^{2n}\,\varphi\circ F^{n}1_{\{\varphi_{n+1}\circ F^{n}>t\}}\,d\mu
=2C2γ1n|v|γ|w|YφφFn1{φn+1>t}𝑑μ.absent2subscript𝐶2superscriptsubscript𝛾1𝑛subscript𝑣𝛾subscript𝑤subscript𝑌𝜑𝜑superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡differential-d𝜇\displaystyle=2C_{2}\gamma_{1}^{n}|v|_{\gamma}|w|_{\infty}\int_{Y}\varphi\,\varphi\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu.

The result follows from Propositions 3.4(b) and 4.13. ∎

4.4 Estimates for C^j,ksubscript^𝐶𝑗𝑘{\widehat{C}}_{j,k}

For the moment, we suppose that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) where β>1𝛽1\beta>1, and that q𝑞q, η𝜂\eta, γ1subscript𝛾1\gamma_{1}, θ𝜃\theta are as in Subsection 4.1. First, we estimate the inverse Laplace transform W¯k(t):Y¯:subscript¯𝑊𝑘𝑡¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(t):{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} associated to W^k(s):Y:subscript^𝑊𝑘𝑠𝑌{\widehat{W}}_{k}(s):Y\to{\mathbb{C}}.

Proposition 4.16

There is a constant C>0𝐶0C>0 such that

|W¯k(t)|1C(k+1)β+1γ1kwγ(t+1)qfor all wγ(Yφ)k0t>0.subscriptsubscript¯𝑊𝑘𝑡1𝐶superscript𝑘1𝛽1superscriptsubscript𝛾1𝑘subscriptnorm𝑤𝛾superscript𝑡1𝑞for all wγ(Yφ)k0t>0.|{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(t)|_{1}\leq C(k+1)^{\beta+1}\gamma_{1}^{k}\|w\|_{\gamma}\,(t+1)^{-q}\quad\text{for all $w\in{\mathcal{H}}_{\gamma}(Y^{\varphi})$, $k\geq 0$, $t>0$.}
Proof.

For all k0𝑘0k\geq 0,

W^k(s)(y)=esφk(y)Δkw^(s)(y)subscript^𝑊𝑘𝑠𝑦superscript𝑒𝑠subscript𝜑𝑘𝑦subscriptΔ𝑘^𝑤𝑠𝑦\displaystyle{\widehat{W}}_{k}(s)(y)=e^{-s\varphi_{k}(y)}\Delta_{k}{\widehat{w}}(s)(y) =0φ(Fky)es(φk(y)+u)Δ~kw(y,u)𝑑uabsentsuperscriptsubscript0𝜑superscript𝐹𝑘𝑦superscript𝑒𝑠subscript𝜑𝑘𝑦𝑢subscript~Δ𝑘𝑤𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{k}y)}e^{-s(\varphi_{k}(y)+u)}\widetilde{\Delta}_{k}w(y,u)\,du
=φk(y)φk+1(y)estΔ~kw(y,tφk(y))𝑑t.absentsuperscriptsubscriptsubscript𝜑𝑘𝑦subscript𝜑𝑘1𝑦superscript𝑒𝑠𝑡subscript~Δ𝑘𝑤𝑦𝑡subscript𝜑𝑘𝑦differential-d𝑡\displaystyle=\int_{\varphi_{k}(y)}^{\varphi_{k+1}(y)}e^{-st}\widetilde{\Delta}_{k}w(y,t-\varphi_{k}(y))\,dt.

Hence

Wk(t)(y)=1{φk(y)<t<φk+1(y)}Δ~kw(y,tφk(y)),subscript𝑊𝑘𝑡𝑦subscript1subscript𝜑𝑘𝑦𝑡subscript𝜑𝑘1𝑦subscript~Δ𝑘𝑤𝑦𝑡subscript𝜑𝑘𝑦W_{k}(t)(y)=1_{\{\varphi_{k}(y)<t<\varphi_{k+1}(y)\}}\widetilde{\Delta}_{k}w(y,t-\varphi_{k}(y)),

and |Wk(t)|2C2γ1k1wγ(φFk)η1{φk+1>t}subscript𝑊𝑘𝑡2subscript𝐶2superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾superscript𝜑superscript𝐹𝑘𝜂subscript1subscript𝜑𝑘1𝑡|W_{k}(t)|\leq 2C_{2}\gamma_{1}^{k-1}\|w\|_{\gamma}(\varphi\circ F^{k})^{\eta}1_{\{\varphi_{k+1}>t\}} by Proposition 4.12(a). It follows that

|W¯k(t)|1subscriptsubscript¯𝑊𝑘𝑡1\displaystyle|{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(t)|_{1} =|Wk(t)|12C2(k+1)γ1k1wγYφη1{φ>t/(k+1)}𝑑μabsentsubscriptsubscript𝑊𝑘𝑡12subscript𝐶2𝑘1superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾subscript𝑌superscript𝜑𝜂subscript1𝜑𝑡𝑘1differential-d𝜇\displaystyle=|W_{k}(t)|_{1}\leq\textstyle 2C_{2}(k+1)\gamma_{1}^{k-1}\|w\|_{\gamma}\int_{Y}\varphi^{\eta}1_{\{\varphi>t/(k+1)\}}\,d\mu
(k+1)β+1ηγ1k1wγ(t+1)(βη)(k+1)β+1γ1k1wγ(t+1)q,much-less-thanabsentsuperscript𝑘1𝛽1𝜂superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾superscript𝑡1𝛽𝜂superscript𝑘1𝛽1superscriptsubscript𝛾1𝑘1subscriptnorm𝑤𝛾superscript𝑡1𝑞\displaystyle\ll(k+1)^{\beta+1-\eta}\gamma_{1}^{k-1}\|w\|_{\gamma}\,(t+1)^{-(\beta-\eta)}\leq(k+1)^{\beta+1}\gamma_{1}^{k-1}\|w\|_{\gamma}\,(t+1)^{-q},

by Proposition 3.4. ∎

Proposition 4.17

There exists C>0𝐶0C>0 such that

(R^)(q)(s)θCq(|s|+1)for all s=a+ib with a[0,1] and all 1.subscriptnormsuperscriptsuperscript^𝑅𝑞𝑠𝜃𝐶superscript𝑞𝑠1for all s=a+ib with a[0,1] and all 1\|({\widehat{R}}^{\ell})^{(q)}(s)\|_{\theta}\leq C\ell^{q}(|s|+1)\quad\text{for all $s=a+ib\in{\mathbb{C}}$ with $a\in[0,1]$ and all $\ell\geq 1$}.
Proof.

By Proposition 4.6, there exists a constant M>0𝑀0M>0 such that R^(p)(s)bMsubscriptnormsuperscript^𝑅𝑝𝑠𝑏𝑀\|{\widehat{R}}^{(p)}(s)\|_{b}\leq M for all pq𝑝𝑞p\leq q. Also R^(s)nbM1subscriptnorm^𝑅superscript𝑠𝑛𝑏subscript𝑀1\|{\widehat{R}}(s)^{n}\|_{b}\leq M_{1} by (4.1).

For q1𝑞1q\geq 1, note that (R^)(q)superscriptsuperscript^𝑅𝑞({\widehat{R}}^{\ell})^{(q)} consists of qsuperscript𝑞\ell^{q} terms (counting repetitions) of the form

R^n1R^(p1)R^nkR^(pk)R^nk+1,superscript^𝑅subscript𝑛1superscript^𝑅subscript𝑝1superscript^𝑅subscript𝑛𝑘superscript^𝑅subscript𝑝𝑘superscript^𝑅subscript𝑛𝑘1{\widehat{R}}^{n_{1}}{\widehat{R}}^{(p_{1})}\cdots{\widehat{R}}^{n_{k}}{\widehat{R}}^{(p_{k})}{\widehat{R}}^{n_{k+1}},

where ni0subscript𝑛𝑖0n_{i}\geq 0, 1piq1subscript𝑝𝑖𝑞1\leq p_{i}\leq q, n1++nk+1+k=subscript𝑛1subscript𝑛𝑘1𝑘n_{1}+\dots+n_{k+1}+k=\ell, p1++pk=qsubscript𝑝1subscript𝑝𝑘𝑞p_{1}+\dots+p_{k}=q. Since kq𝑘𝑞k\leq q,

R^n1R^(p1)R^nkR^(pk)R^nk+1bM1q+1Mq.subscriptnormsuperscript^𝑅subscript𝑛1superscript^𝑅subscript𝑝1superscript^𝑅subscript𝑛𝑘superscript^𝑅subscript𝑝𝑘superscript^𝑅subscript𝑛𝑘1𝑏superscriptsubscript𝑀1𝑞1superscript𝑀𝑞\|{\widehat{R}}^{n_{1}}{\widehat{R}}^{(p_{1})}\cdots{\widehat{R}}^{n_{k}}{\widehat{R}}^{(p_{k})}{\widehat{R}}^{n_{k+1}}\|_{b}\leq M_{1}^{q+1}M^{q}.

Hence (R^)(q)(s)θ(M0+1)(|s|+1)(R^)(q)(s)bq(|s|+1)subscriptnormsuperscriptsuperscript^𝑅𝑞𝑠𝜃subscript𝑀01𝑠1subscriptnormsuperscriptsuperscript^𝑅𝑞𝑠𝑏much-less-thansuperscript𝑞𝑠1\|({\widehat{R}}^{\ell})^{(q)}(s)\|_{\theta}\leq(M_{0}+1)(|s|+1)\|({\widehat{R}}^{\ell})^{(q)}(s)\|_{b}\ll\ell^{q}(|s|+1). ∎

Proposition 4.18

Let vγ(Yφ)𝑣subscript𝛾superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma}(Y^{\varphi}). Define I0(s)=Y0φ(y)es(φ(y)u)v(y,u)𝑑u𝑑μsubscript𝐼0𝑠subscript𝑌superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝜑𝑦𝑢𝑣𝑦𝑢differential-d𝑢differential-d𝜇I_{0}(s)=\int_{Y}\int_{0}^{\varphi(y)}e^{-s(\varphi(y)-u)}v(y,u)\,du\,d\mu. Then

j=0YV^j𝑑μ=I0on ¯.superscriptsubscript𝑗0subscript𝑌subscript^𝑉𝑗differential-d𝜇subscript𝐼0on ¯.\textstyle\sum_{j=0}^{\infty}\int_{Y}{\widehat{V}}_{j}\,d\mu=I_{0}\quad\text{on ${\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}$.}
Proof.

For j1𝑗1j\geq 1,

YV^j(s)𝑑μsubscript𝑌subscript^𝑉𝑗𝑠differential-d𝜇\displaystyle\int_{Y}{\widehat{V}}_{j}(s)\,d\mu =Y0φ(Fjy)es(φ(Fjy)u)(v(Fjπy,u)v(Fj1πFy,u)dudμ\displaystyle=\int_{Y}\int_{0}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}(v(F^{j}\pi y,u)-v(F^{j-1}\pi Fy,u)\,du\,d\mu
=Y0φ(Fjy)es(φ(Fjy)u)v(Fjπy,u)𝑑u𝑑μabsentsubscript𝑌superscriptsubscript0𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢𝑣superscript𝐹𝑗𝜋𝑦𝑢differential-d𝑢differential-d𝜇\displaystyle=\int_{Y}\int_{0}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}v(F^{j}\pi y,u)\,du\,d\mu
Y0φ(Fj1y)es(φ(Fj1y)u)v(Fj1πy,u)𝑑u𝑑μ,subscript𝑌superscriptsubscript0𝜑superscript𝐹𝑗1𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗1𝑦𝑢𝑣superscript𝐹𝑗1𝜋𝑦𝑢differential-d𝑢differential-d𝜇\displaystyle\qquad\qquad-\int_{Y}\int_{0}^{\varphi(F^{j-1}y)}e^{-s(\varphi(F^{j-1}y)-u)}v(F^{j-1}\pi y,u)\,du\,d\mu,

while YV^0(s)𝑑μ=Y0φ(y)es(φ(y)u)v(πy,u)𝑑u𝑑μsubscript𝑌subscript^𝑉0𝑠differential-d𝜇subscript𝑌superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝜑𝑦𝑢𝑣𝜋𝑦𝑢differential-d𝑢differential-d𝜇\int_{Y}{\widehat{V}}_{0}(s)\,d\mu=\int_{Y}\int_{0}^{\varphi(y)}e^{-s(\varphi(y)-u)}v(\pi y,u)\,du\,d\mu. Hence

j=0JYV^j(s)𝑑μsuperscriptsubscript𝑗0𝐽subscript𝑌subscript^𝑉𝑗𝑠differential-d𝜇\displaystyle\sum_{j=0}^{J}\int_{Y}{\widehat{V}}_{j}(s)\,d\mu =Y0φ(FJy)es(φ(FJy)u)v(FJπy,u)𝑑u𝑑μabsentsubscript𝑌superscriptsubscript0𝜑superscript𝐹𝐽𝑦superscript𝑒𝑠𝜑superscript𝐹𝐽𝑦𝑢𝑣superscript𝐹𝐽𝜋𝑦𝑢differential-d𝑢differential-d𝜇\displaystyle=\int_{Y}\int_{0}^{\varphi(F^{J}y)}e^{-s(\varphi(F^{J}y)-u)}v(F^{J}\pi y,u)\,du\,d\mu
=ZJ(s)+Y0φ(FJy)es(φ(FJy)u)v(FJy,u)𝑑u𝑑μ=ZJ(s)+I0(s),absentsubscript𝑍𝐽𝑠subscript𝑌superscriptsubscript0𝜑superscript𝐹𝐽𝑦superscript𝑒𝑠𝜑superscript𝐹𝐽𝑦𝑢𝑣superscript𝐹𝐽𝑦𝑢differential-d𝑢differential-d𝜇subscript𝑍𝐽𝑠subscript𝐼0𝑠\displaystyle=Z_{J}(s)+\int_{Y}\int_{0}^{\varphi(F^{J}y)}e^{-s(\varphi(F^{J}y)-u)}v(F^{J}y,u)\,du\,d\mu=Z_{J}(s)+I_{0}(s),

where

ZJ(s)=Y0φ(FJy)es(φ(FJy)u)(v(FJπy,u)v(FJy,u))𝑑u𝑑μ.subscript𝑍𝐽𝑠subscript𝑌superscriptsubscript0𝜑superscript𝐹𝐽𝑦superscript𝑒𝑠𝜑superscript𝐹𝐽𝑦𝑢𝑣superscript𝐹𝐽𝜋𝑦𝑢𝑣superscript𝐹𝐽𝑦𝑢differential-d𝑢differential-d𝜇Z_{J}(s)=\int_{Y}\int_{0}^{\varphi(F^{J}y)}e^{-s(\varphi(F^{J}y)-u)}(v(F^{J}\pi y,u)-v(F^{J}y,u))\,du\,d\mu.

By (3.1),

|v(FJπy,u)v(FJy,u)||v|γφ(FJy)d(FJπy,FJy)C2γJ|v|γφ(FJy).𝑣superscript𝐹𝐽𝜋𝑦𝑢𝑣superscript𝐹𝐽𝑦𝑢subscript𝑣𝛾𝜑superscript𝐹𝐽𝑦𝑑superscript𝐹𝐽𝜋𝑦superscript𝐹𝐽𝑦subscript𝐶2superscript𝛾𝐽subscript𝑣𝛾𝜑superscript𝐹𝐽𝑦|v(F^{J}\pi y,u)-v(F^{J}y,u)|\leq|v|_{\gamma}\,\varphi(F^{J}y)d(F^{J}\pi y,F^{J}y)\leq C_{2}\gamma^{J}|v|_{\gamma}\,\varphi(F^{J}y).

Also, |v(FJπy,u)v(FJy,u)|2|v|𝑣superscript𝐹𝐽𝜋𝑦𝑢𝑣superscript𝐹𝐽𝑦𝑢2subscript𝑣|v(F^{J}\pi y,u)-v(F^{J}y,u)|\leq 2|v|_{\infty}, so

|v(FJπy,u)v(FJy,u)|2C2γ1Jvγφ(FJy)η.𝑣superscript𝐹𝐽𝜋𝑦𝑢𝑣superscript𝐹𝐽𝑦𝑢2subscript𝐶2superscriptsubscript𝛾1𝐽subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝐽𝑦𝜂|v(F^{J}\pi y,u)-v(F^{J}y,u)|\leq 2C_{2}\gamma_{1}^{J}\|v\|_{\gamma}\,\varphi(F^{J}y)^{\eta}.

Hence |ZJ(s)|2C2γ1JvγY(φFJ)1+η𝑑μ=2C2γ1JvγYφ1+η𝑑μ0subscript𝑍𝐽𝑠2subscript𝐶2superscriptsubscript𝛾1𝐽subscriptnorm𝑣𝛾subscript𝑌superscript𝜑superscript𝐹𝐽1𝜂differential-d𝜇2subscript𝐶2superscriptsubscript𝛾1𝐽subscriptnorm𝑣𝛾subscript𝑌superscript𝜑1𝜂differential-d𝜇0|Z_{J}(s)|\leq 2C_{2}\gamma_{1}^{J}\|v\|_{\gamma}\int_{Y}(\varphi\circ F^{J})^{1+\eta}\,d\mu=2C_{2}\gamma_{1}^{J}\|v\|_{\gamma}\int_{Y}\varphi^{1+\eta}\,d\mu\to 0 as J𝐽J\to\infty. ∎

From now on, we specialize to the rapid mixing case, so q𝑞q and β𝛽\beta are arbitrarily large and all functions previously regarded as Cqsuperscript𝐶𝑞C^{q} are now Csuperscript𝐶C^{\infty}. Note that

min{γ1j,γ1s(y,y)j}γ113jγ113s(y,y)γ113jθs(y,y).superscriptsubscript𝛾1𝑗superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗superscriptsubscript𝛾113𝑗superscriptsubscript𝛾113𝑠𝑦superscript𝑦superscriptsubscript𝛾113𝑗superscript𝜃𝑠𝑦superscript𝑦\min\{\gamma_{1}^{j},\gamma_{1}^{s(y,y^{\prime})-j}\}\leq\gamma_{1}^{\frac{1}{3}j}\gamma_{1}^{\frac{1}{3}s(y,y^{\prime})}\leq\gamma_{1}^{\frac{1}{3}j}\theta^{s(y,y^{\prime})}. (4.2)
Proposition 4.19

For each r𝑟r\in{\mathbb{N}} there exists C>0𝐶0C>0 such that

Rj+1V¯j(r)(s)θC(|s|+1)γ1j/3vγ for all vγ(Yφ)s¯j0.subscriptnormsuperscript𝑅𝑗1superscriptsubscript¯𝑉𝑗𝑟𝑠𝜃𝐶𝑠1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾 for all vγ(Yφ)s¯j0.\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)\|_{\theta}\leq C(|s|+1)\gamma_{1}^{j/3}\|v\|_{\gamma}\quad\text{ for all $v\in{\mathcal{H}}_{\gamma}(Y^{\varphi})$, $s\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}$, $j\geq 0$.}
Proof.

For j0𝑗0j\geq 0,

V^j(s)(y)=esφ(Fjy)Δjvs(y)=0φ(Fjy)es(φ(Fjy)u)Δ~jv(y,u)𝑑u.subscript^𝑉𝑗𝑠𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦subscriptΔ𝑗subscript𝑣𝑠𝑦superscriptsubscript0𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢{\widehat{V}}_{j}(s)(y)=e^{-s\varphi(F^{j}y)}\Delta_{j}v_{s}(y)=\int_{0}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}\widetilde{\Delta}_{j}v(y,u)\,du.

Hence

V^j(r)(s)(y)=(1)r0φ(Fjy)es(φ(Fjy)u)(φ(Fjy)u)rΔ~jv(y,u)𝑑u.superscriptsubscript^𝑉𝑗𝑟𝑠𝑦superscript1𝑟superscriptsubscript0𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢superscript𝜑superscript𝐹𝑗𝑦𝑢𝑟subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢{\widehat{V}}_{j}^{(r)}(s)(y)=(-1)^{r}\int_{0}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}(\varphi(F^{j}y)-u)^{r}\widetilde{\Delta}_{j}v(y,u)\,du.

By Proposition 4.12(a), |Δ~jv(y,u)|2C2γ1j1vγφ(Fjy)ηsubscript~Δ𝑗𝑣𝑦𝑢2subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂|\widetilde{\Delta}_{j}v(y,u)|\leq 2C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\varphi(F^{j}y)^{\eta}. Hence |V^j(r)(s)|2C2γ1j1vγφr+2Fjsuperscriptsubscript^𝑉𝑗𝑟𝑠2subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾superscript𝜑𝑟2superscript𝐹𝑗|{\widehat{V}}_{j}^{(r)}(s)|\leq 2C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\,\varphi^{r+2}\circ F^{j}.

Fix a (j+1)𝑗1(j+1)-cylinder d𝑑d for the Gibbs-Markov map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. Since F¯jdsuperscript¯𝐹𝑗𝑑\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d is a partition element,

|1dV¯j(r)(s)|C2γ1j1vγ|1F¯jdφ|r+2(2C1)r+2C2γ1j1vγinfF¯jdφr+2.subscriptsubscript1𝑑superscriptsubscript¯𝑉𝑗𝑟𝑠subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾superscriptsubscriptsubscript1superscript¯𝐹𝑗𝑑𝜑𝑟2superscript2subscript𝐶1𝑟2subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝑟2|1_{d}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)|_{\infty}\leq C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\,|1_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}\,\varphi|_{\infty}^{r+2}\leq(2C_{1})^{r+2}C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{r+2}. (4.3)

Let y,yd𝑦superscript𝑦𝑑y,y^{\prime}\in d with φ(F¯jy)φ(F¯jy)𝜑superscript¯𝐹𝑗𝑦𝜑superscript¯𝐹𝑗superscript𝑦\varphi(\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}y)\geq\varphi(\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}y^{\prime}). Then

V¯j(r)(s)(y)V¯j(r)(s)(y)=(1)r(I1+I2+I3+I4),superscriptsubscript¯𝑉𝑗𝑟𝑠𝑦superscriptsubscript¯𝑉𝑗𝑟𝑠superscript𝑦superscript1𝑟subscript𝐼1subscript𝐼2subscript𝐼3subscript𝐼4{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y)-{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y^{\prime})=(-1)^{r}(I_{1}+I_{2}+I_{3}+I_{4}),

where

I1subscript𝐼1\displaystyle I_{1} =φ(Fjy)φ(Fjy)es(φ(Fjy)u)(φ(Fjy)u)rΔ~jv(y,u)𝑑u,absentsuperscriptsubscript𝜑superscript𝐹𝑗superscript𝑦𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢superscript𝜑superscript𝐹𝑗𝑦𝑢𝑟subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢\displaystyle=\int_{\varphi(F^{j}y^{\prime})}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}(\varphi(F^{j}y)-u)^{r}\widetilde{\Delta}_{j}v(y,u)\,du,
I2subscript𝐼2\displaystyle I_{2} =0φ(Fjy){es(φ(Fjy)u)es(φ(Fjy)u)}(φ(Fjy)u)rΔ~jv(y,u)𝑑u,absentsuperscriptsubscript0𝜑superscript𝐹𝑗superscript𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢superscript𝑒𝑠𝜑superscript𝐹𝑗superscript𝑦𝑢superscript𝜑superscript𝐹𝑗𝑦𝑢𝑟subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{j}y^{\prime})}\{e^{-s(\varphi(F^{j}y)-u)}-e^{-s(\varphi(F^{j}y^{\prime})-u)}\}(\varphi(F^{j}y)-u)^{r}\widetilde{\Delta}_{j}v(y,u)\,du,
I3subscript𝐼3\displaystyle I_{3} =0φ(Fjy)es(φ(Fjy)u){(φ(Fjy)u)r(φ(Fjy)u)r}Δ~jv(y,u)𝑑u,absentsuperscriptsubscript0𝜑superscript𝐹𝑗superscript𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗superscript𝑦𝑢superscript𝜑superscript𝐹𝑗𝑦𝑢𝑟superscript𝜑superscript𝐹𝑗superscript𝑦𝑢𝑟subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{j}y^{\prime})}e^{-s(\varphi(F^{j}y^{\prime})-u)}\{(\varphi(F^{j}y)-u)^{r}-(\varphi(F^{j}y^{\prime})-u)^{r}\}\widetilde{\Delta}_{j}v(y,u)\,du,
I4subscript𝐼4\displaystyle I_{4} =0φ(Fjy)es(φ(Fjy)u)(φ(Fjy)u)r{Δ~jv(y,u)Δ~jv(y,u)}𝑑u.absentsuperscriptsubscript0𝜑superscript𝐹𝑗superscript𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗superscript𝑦𝑢superscript𝜑superscript𝐹𝑗superscript𝑦𝑢𝑟subscript~Δ𝑗𝑣𝑦𝑢subscript~Δ𝑗𝑣superscript𝑦𝑢differential-d𝑢\displaystyle=\int_{0}^{\varphi(F^{j}y^{\prime})}e^{-s(\varphi(F^{j}y^{\prime})-u)}(\varphi(F^{j}y^{\prime})-u)^{r}\{\widetilde{\Delta}_{j}v(y,u)-\widetilde{\Delta}_{j}v(y^{\prime},u)\}\,du.

By (3.3),

|φ(Fjy)φ(Fjy)|C1infF¯jdφγs(Fjy,Fjy)=C1γs(y,y)jinfF¯jdφ.𝜑superscript𝐹𝑗𝑦𝜑superscript𝐹𝑗superscript𝑦subscript𝐶1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝛾𝑠superscript𝐹𝑗𝑦superscript𝐹𝑗superscript𝑦subscript𝐶1superscript𝛾𝑠𝑦superscript𝑦𝑗subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑|\varphi(F^{j}y)-\varphi(F^{j}y^{\prime})|\leq C_{1}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi\,\gamma^{s(F^{j}y,F^{j}y^{\prime})}=C_{1}\gamma^{s(y,y^{\prime})-j}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi.

Hence by Proposition 4.12(a,b),

|V¯j(r)(s)(y)V¯j(r)(s)(y)|(|s|+1)γ1s(y,y)jvγinfF¯jdφr+3.much-less-thansuperscriptsubscript¯𝑉𝑗𝑟𝑠𝑦superscriptsubscript¯𝑉𝑗𝑟𝑠superscript𝑦𝑠1superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝑟3\displaystyle|{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y)-{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y^{\prime})|\ll(|s|+1)\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{r+3}.
At the same time, the supnorm estimate (4.3) yields
|V¯j(r)(s)(y)V¯j(r)(s)(y)|γ1jvγinfF¯jdφr+3.much-less-thansuperscriptsubscript¯𝑉𝑗𝑟𝑠𝑦superscriptsubscript¯𝑉𝑗𝑟𝑠superscript𝑦superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝑟3\displaystyle|{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y)-{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y^{\prime})|\ll\gamma_{1}^{j}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{r+3}.
Combining these estimates and using (4.2) we obtain that
|V¯j(r)(s)(y)V¯j(r)(s)(y)|(|s|+1)γ1j/3θs(y,y)vγinfF¯jdφr+3.much-less-thansuperscriptsubscript¯𝑉𝑗𝑟𝑠𝑦superscriptsubscript¯𝑉𝑗𝑟𝑠superscript𝑦𝑠1superscriptsubscript𝛾1𝑗3superscript𝜃𝑠𝑦superscript𝑦subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝑟3\displaystyle|{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y)-{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)(y^{\prime})|\ll(|s|+1)\gamma_{1}^{j/3}\theta^{s(y,y^{\prime})}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{r+3}.

In other words,

|1dV¯j(r)(s)|θ(|s|+1)γ1j/3vγinfF¯jdφr+3.much-less-thansubscriptsubscript1𝑑superscriptsubscript¯𝑉𝑗𝑟𝑠𝜃𝑠1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝑟3|1_{d}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)|_{\theta}\ll(|s|+1)\gamma_{1}^{j/3}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{r+3}.

Using this and (4.3), it follows by Proposition 4.4 that

Rj+1V¯j(r)(s)θsubscriptnormsuperscript𝑅𝑗1superscriptsubscript¯𝑉𝑗𝑟𝑠𝜃\displaystyle\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}_{j}^{(r)}(s)\|_{\theta} (|s|+1)γ1j/3vγdμ¯(d)infdφr+3F¯jmuch-less-thanabsent𝑠1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscript𝑑¯𝜇𝑑subscriptinfimum𝑑superscript𝜑𝑟3superscript¯𝐹𝑗\displaystyle\ll(|s|+1)\gamma_{1}^{j/3}\|v\|_{\gamma}{\textstyle\sum_{d}}\,\bar{\mu}(d){\textstyle\inf_{d}}\,\varphi^{r+3}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}
(|s|+1)γ1j/3vγY¯φr+3F¯j𝑑μ¯=(|s|+1)γ1j/3vγYφr+3𝑑μ,absent𝑠1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscript¯𝑌superscript𝜑𝑟3superscript¯𝐹𝑗differential-d¯𝜇𝑠1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscript𝑌superscript𝜑𝑟3differential-d𝜇\displaystyle\leq(|s|+1)\gamma_{1}^{j/3}\|v\|_{\gamma}{\textstyle\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}}\varphi^{r+3}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}\,d\bar{\mu}=(|s|+1)\gamma_{1}^{j/3}\|v\|_{\gamma}{\textstyle\int_{Y}}\varphi^{r+3}\,d\mu,

completing the proof. ∎

Define Dj,=R^T^Rj+1V¯jsubscript𝐷𝑗superscript^𝑅^𝑇superscript𝑅𝑗1subscript¯𝑉𝑗D_{j,\ell}={\widehat{R}}^{\ell}{\widehat{T}}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}, j,0𝑗0j,\ell\geq 0. Let δ𝛿\delta and λ𝜆\lambda be as in Proposition 4.9, and recall that δ=¯Bδ(0)subscript𝛿¯subscript𝐵𝛿0{\mathbb{H}}_{\delta}={\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\cap B_{\delta}(0).

Proposition 4.20

For each r𝑟r\in{\mathbb{N}}, there exists α,C>0𝛼𝐶0\alpha,\,C>0 such that for all vγ(Yφ)𝑣subscript𝛾superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma}(Y^{\varphi}), j,0𝑗0j,\ell\geq 0, and all s=a+ib𝑠𝑎𝑖𝑏s=a+ib\in{\mathbb{C}} with a[0,1]𝑎01a\in[0,1],

  • (a)

    |Dj,(r)(s)|C(+1)rγ1j/3(|b|+1)αvγsubscriptsuperscriptsubscript𝐷𝑗𝑟𝑠𝐶superscript1𝑟superscriptsubscript𝛾1𝑗3superscript𝑏1𝛼subscriptnorm𝑣𝛾|D_{j,\ell}^{(r)}(s)|_{\infty}\leq C(\ell+1)^{r}\gamma_{1}^{j/3}(|b|+1)^{\alpha}\|v\|_{\gamma} for sδ𝑠subscript𝛿s\not\in{\mathbb{H}}_{\delta},

  • (b)

    |drdsr{Dj,(s)(1λ(s))1YV^j(s)𝑑μ}|C(+1)r+1γ1j/3vγsubscriptsuperscript𝑑𝑟𝑑superscript𝑠𝑟subscript𝐷𝑗𝑠superscript1𝜆𝑠1subscript𝑌subscript^𝑉𝑗𝑠differential-d𝜇𝐶superscript1𝑟1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾|\frac{d^{r}}{ds^{r}}\{D_{j,\ell}(s)-(1-\lambda(s))^{-1}\int_{Y}{\widehat{V}}_{j}(s)\,d\mu\}|_{\infty}\leq C(\ell+1)^{r+1}\gamma_{1}^{j/3}\|v\|_{\gamma} for sδ𝑠subscript𝛿s\in{\mathbb{H}}_{\delta}.

Proof.

Let p𝑝p\in{\mathbb{N}}, pr𝑝𝑟p\leq r. By Propositions 4.17 and 4.19, Rj+1V¯j(p)(s)θγ1j/3(|b|+1)vγmuch-less-thansubscriptnormsuperscript𝑅𝑗1superscriptsubscript¯𝑉𝑗𝑝𝑠𝜃superscriptsubscript𝛾1𝑗3𝑏1subscriptnorm𝑣𝛾\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}^{(p)}(s)\|_{\theta}\ll\gamma_{1}^{j/3}(|b|+1)\|v\|_{\gamma}, and (R^)(p)(s)θ(+1)r(|b|+1)much-less-thansubscriptnormsuperscriptsuperscript^𝑅𝑝𝑠𝜃superscript1𝑟𝑏1\|({\widehat{R}}^{\ell})^{(p)}(s)\|_{\theta}\ll(\ell+1)^{r}(|b|+1).

For sδ𝑠subscript𝛿s\not\in{\mathbb{H}}_{\delta}, it follows from Proposition 4.8 that T^(p)(s)θ(|b|+1)αmuch-less-thansubscriptnormsuperscript^𝑇𝑝𝑠𝜃superscript𝑏1𝛼\|{\widehat{T}}^{(p)}(s)\|_{\theta}\ll(|b|+1)^{\alpha} for some α>0𝛼0\alpha>0. Combining these estimates,

|(R^T^Rj+1V¯j)(r)(s)|subscriptsuperscriptsuperscript^𝑅^𝑇superscript𝑅𝑗1subscript¯𝑉𝑗𝑟𝑠\displaystyle|({\widehat{R}}^{\ell}{\widehat{T}}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j})^{(r)}(s)|_{\infty} (R^T^Rj+1V¯j)(r)(s)θ(+1)rγ1j/3(|b|+1)α+2vγ,absentsubscriptnormsuperscriptsuperscript^𝑅^𝑇superscript𝑅𝑗1subscript¯𝑉𝑗𝑟𝑠𝜃much-less-thansuperscript1𝑟superscriptsubscript𝛾1𝑗3superscript𝑏1𝛼2subscriptnorm𝑣𝛾\displaystyle\leq\|({\widehat{R}}^{\ell}{\widehat{T}}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j})^{(r)}(s)\|_{\theta}\ll(\ell+1)^{r}\gamma_{1}^{j/3}(|b|+1)^{\alpha+2}\|v\|_{\gamma},

completing the proof of (a).

Next, suppose that sδ𝑠subscript𝛿s\in{\mathbb{H}}_{\delta}. By Proposition 4.9, R^=λP+R^Q^𝑅𝜆𝑃^𝑅𝑄{\widehat{R}}=\lambda P+{\widehat{R}}Q where P(s)𝑃𝑠P(s) is the spectral projection corresponding to λ(s)𝜆𝑠\lambda(s) and Q(s)=IP(s)𝑄𝑠𝐼𝑃𝑠Q(s)=I-P(s). By Proposition 4.9, λ(s)𝜆𝑠\lambda(s) is a Csuperscript𝐶C^{\infty} family of isolated eigenvalues with λ(0)=1𝜆01\lambda(0)=1, λ(0)0superscript𝜆00\lambda^{\prime}(0)\neq 0 and |λ(s)|1𝜆𝑠1|\lambda(s)|\leq 1, and P(s)𝑃𝑠P(s) is a Csuperscript𝐶C^{\infty} family of operators on θ(Y¯)subscript𝜃¯𝑌{\mathcal{F}}_{\theta}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}) with P(0)v=Y¯v𝑑μ¯𝑃0𝑣subscript¯𝑌𝑣differential-d¯𝜇P(0)v=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}v\,d\bar{\mu}. Also

T^=(1λ)1P+Q1on δ{0},^𝑇superscript1𝜆1𝑃subscript𝑄1on δ{0}{\widehat{T}}=(1-\lambda)^{-1}P+Q_{1}\quad\text{on ${\mathbb{H}}_{\delta}\setminus\{0\}$},

where Q1=T^Qsubscript𝑄1^𝑇𝑄Q_{1}={\widehat{T}}Q is Csuperscript𝐶C^{\infty} on δsubscript𝛿{\mathbb{H}}_{\delta}. Hence

R^T^superscript^𝑅^𝑇\displaystyle{\widehat{R}}^{\ell}{\widehat{T}} =(1λ)1λP+R^Q1=(1λ)1λP(0)+λQ2+R^Q1on δ{0},formulae-sequenceabsentsuperscript1𝜆1superscript𝜆𝑃superscript^𝑅subscript𝑄1superscript1𝜆1superscript𝜆𝑃0superscript𝜆subscript𝑄2superscript^𝑅subscript𝑄1on δ{0},\displaystyle=(1-\lambda)^{-1}\lambda^{\ell}P+{\widehat{R}}^{\ell}Q_{1}=(1-\lambda)^{-1}\lambda^{\ell}P(0)+\lambda^{\ell}Q_{2}+{\widehat{R}}^{\ell}Q_{1}\quad\text{on ${\mathbb{H}}_{\delta}\setminus\{0\}$,}

where Q2=(1λ)1(PP(0))subscript𝑄2superscript1𝜆1𝑃𝑃0Q_{2}=(1-\lambda)^{-1}(P-P(0)) is Csuperscript𝐶C^{\infty} on δsubscript𝛿{\mathbb{H}}_{\delta}. Also, (1λ)1λ=(1λ)1(λ1++1)superscript1𝜆1superscript𝜆superscript1𝜆1superscript𝜆11(1-\lambda)^{-1}\lambda^{\ell}=(1-\lambda)^{-1}-(\lambda^{\ell-1}+\dots+1), so

Dj,(1λ)1P(0)Rj+1V¯j=Qj,on δ,subscript𝐷𝑗superscript1𝜆1𝑃0superscript𝑅𝑗1subscript¯𝑉𝑗subscript𝑄𝑗on δ,D_{j,\ell}-(1-\lambda)^{-1}P(0)R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}=Q_{j,\ell}\quad\text{on ${\mathbb{H}}_{\delta}$,}

where

Qj,=((λ1++1)P(0)+λQ2+R^Q1)Rj+1V¯j.subscript𝑄𝑗superscript𝜆11𝑃0superscript𝜆subscript𝑄2superscript^𝑅subscript𝑄1superscript𝑅𝑗1subscript¯𝑉𝑗Q_{j,\ell}=\big{(}-(\lambda^{\ell-1}+\dots+1)P(0)+\lambda^{\ell}Q_{2}+{\widehat{R}}^{\ell}Q_{1}\big{)}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}.

It follows from the estimates for Rj+1V¯jsuperscript𝑅𝑗1subscript¯𝑉𝑗R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j} and R^superscript^𝑅{\widehat{R}}^{\ell} that |(R^Q1Rj+1V¯j)(r)(s)|(+1)rγ1j/3vγmuch-less-thansubscriptsuperscriptsuperscript^𝑅subscript𝑄1superscript𝑅𝑗1subscript¯𝑉𝑗𝑟𝑠superscript1𝑟superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾|({\widehat{R}}^{\ell}Q_{1}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j})^{(r)}(s)|_{\infty}\ll(\ell+1)^{r}\gamma_{1}^{j/3}\|v\|_{\gamma} for sδ𝑠subscript𝛿s\in{\mathbb{H}}_{\delta}. Since |λ(s)|1𝜆𝑠1|\lambda(s)|\leq 1, the proof of Proposition 4.17 applies equally to λsuperscript𝜆\lambda^{\ell}, so |Qj,(r)(s)|(+1)r+1γ1j/3vγmuch-less-thansubscriptsuperscriptsubscript𝑄𝑗𝑟𝑠superscript1𝑟1superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾|Q_{j,\ell}^{(r)}(s)|_{\infty}\ll(\ell+1)^{r+1}\gamma_{1}^{j/3}\|v\|_{\gamma} for sδ𝑠subscript𝛿s\in{\mathbb{H}}_{\delta}.

Finally P(0)Rj+1V¯j=Y¯V¯j𝑑μ¯=YV^j𝑑μ𝑃0superscript𝑅𝑗1subscript¯𝑉𝑗subscript¯𝑌subscript¯𝑉𝑗differential-d¯𝜇subscript𝑌subscript^𝑉𝑗differential-d𝜇P(0)R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}\,d\bar{\mu}=\int_{Y}{\widehat{V}}_{j}\,d\mu completing the proof of part (b). ∎

By Lemma 4.11, C^=j,k=0C^j,k^𝐶superscriptsubscript𝑗𝑘0subscript^𝐶𝑗𝑘{\widehat{C}}=\sum_{j,k=0}^{\infty}{\widehat{C}}_{j,k} is analytic on {\mathbb{H}}. As shown in the next result, C^^𝐶{\widehat{C}} extends smoothly to ¯¯{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}.

Corollary 4.21

Assume absence of eigenfunctions, and let r𝑟r\in{\mathbb{N}}. There exists α,C>0𝛼𝐶0\alpha,\,C>0 such that

|C^(r)(s)|C(|b|+1)αvγwγ,superscript^𝐶𝑟𝑠𝐶superscript𝑏1𝛼subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾|{\widehat{C}}^{(r)}(s)|\leq C(|b|+1)^{\alpha}\|v\|_{\gamma}\|w\|_{\gamma},

for all s=a+ib¯𝑠𝑎𝑖𝑏¯s=a+ib\in{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu} with a[0,1]𝑎01a\in[0,1], and all v,wγ(Yφ)𝑣𝑤subscript𝛾superscript𝑌𝜑v,w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}) with Yφv𝑑μφ=0subscriptsuperscript𝑌𝜑𝑣differential-dsuperscript𝜇𝜑0\int_{Y^{\varphi}}v\,d\mu^{\varphi}=0.

Proof.

Let =max{jk1,0}𝑗𝑘10\ell=\max\{j-k-1,0\}. Recall from Lemma 4.11 that C^j,k=Y¯Dj,W¯k𝑑μ¯subscript^𝐶𝑗𝑘subscript¯𝑌subscript𝐷𝑗subscript¯𝑊𝑘differential-d¯𝜇{\widehat{C}}_{j,k}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}D_{j,\ell}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\bar{\mu}. Let p𝑝p\in{\mathbb{N}}, pr𝑝𝑟p\leq r. By Proposition 4.16, |W¯k(t)|1(k+1)p+3γ1kwγ(t+1)(p+2)much-less-thansubscriptsubscript¯𝑊𝑘𝑡1superscript𝑘1𝑝3superscriptsubscript𝛾1𝑘subscriptnorm𝑤𝛾superscript𝑡1𝑝2|{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}(t)|_{1}\ll(k+1)^{p+3}\gamma_{1}^{k}\|w\|_{\gamma}\,(t+1)^{-(p+2)}, so |W¯k(p)(s)|1(k+1)r+3γ1kwγmuch-less-thansubscriptsuperscriptsubscript¯𝑊𝑘𝑝𝑠1superscript𝑘1𝑟3superscriptsubscript𝛾1𝑘subscriptnorm𝑤𝛾|{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}^{(p)}(s)|_{1}\ll(k+1)^{r+3}\gamma_{1}^{k}\|w\|_{\gamma}. Combining this with Proposition 4.20(a),

|C^j,k(r)(s)||b|α(j+1)rγ1j/3(k+1)r+3γ1kvγwγfor |b|δ,much-less-thansuperscriptsubscript^𝐶𝑗𝑘𝑟𝑠superscript𝑏𝛼superscript𝑗1𝑟superscriptsubscript𝛾1𝑗3superscript𝑘1𝑟3superscriptsubscript𝛾1𝑘subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾for |b|δ,|{\widehat{C}}_{j,k}^{(r)}(s)|\ll|b|^{\alpha}(j+1)^{r}\gamma_{1}^{j/3}(k+1)^{r+3}\gamma_{1}^{k}\|v\|_{\gamma}\|w\|_{\gamma}\quad\text{for $|b|\geq\delta$,}

and the proof for |b|δ𝑏𝛿|b|\geq\delta is complete.

For |b|δ𝑏𝛿|b|\leq\delta, we use Proposition 4.18 to write

C^=j,kY{Dj,(1λ)1YV^j𝑑μ}W¯k𝑑μ+(1λ)1I0kYW¯k𝑑μ.^𝐶subscript𝑗𝑘subscript𝑌subscript𝐷𝑗superscript1𝜆1subscript𝑌subscript^𝑉𝑗differential-d𝜇subscript¯𝑊𝑘differential-d𝜇superscript1𝜆1subscript𝐼0subscript𝑘subscript𝑌subscript¯𝑊𝑘differential-d𝜇\textstyle{\widehat{C}}=\sum_{j,k}\int_{Y}\big{\{}D_{j,\ell}-(1-\lambda)^{-1}\int_{Y}{\widehat{V}}_{j}\,d\mu\big{\}}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu+(1-\lambda)^{-1}I_{0}\sum_{k}\int_{Y}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu.

Proposition 4.20(b) takes care of the first term on the right-hand side, and it remains to estimate g=(1λ)1I0𝑔superscript1𝜆1subscript𝐼0g=(1-\lambda)^{-1}I_{0}. Now

I0(0)=Y0φ(y)v(y,u)𝑑u𝑑μ=|φ|1Yφv𝑑μφ=0,subscript𝐼00subscript𝑌superscriptsubscript0𝜑𝑦𝑣𝑦𝑢differential-d𝑢differential-d𝜇subscript𝜑1subscriptsuperscript𝑌𝜑𝑣differential-dsuperscript𝜇𝜑0I_{0}(0)=\int_{Y}\int_{0}^{\varphi(y)}v(y,u)\,du\,d\mu=|\varphi|_{1}\int_{Y^{\varphi}}v\,d\mu^{\varphi}=0, (4.4)

so it follows from Proposition 4.9 that g𝑔g is Csuperscript𝐶C^{\infty} with |g(r)(s)||v|much-less-thansuperscript𝑔𝑟𝑠subscript𝑣|g^{(r)}(s)|\ll|v|_{\infty} on δsubscript𝛿{\mathbb{H}}_{\delta}. ∎

Proof of Theorem 3.1   Recall that β𝛽\beta and q𝑞q can be taken arbitrarily large. Hence it follows from Proposition 4.3 that sup¯|J^0(r)||v||w|much-less-thansubscriptsupremum¯superscriptsubscript^𝐽0𝑟subscript𝑣subscript𝑤\sup_{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}|{\widehat{J}}_{0}^{(r)}|\ll|v|_{\infty}|w|_{\infty} for all r𝑟r\in{\mathbb{N}}. Similarly, by Propositions 4.14 and 4.15, sup¯|A^n(r)|nr+3γ1n|v||w|γmuch-less-thansubscriptsupremum¯superscriptsubscript^𝐴𝑛𝑟superscript𝑛𝑟3superscriptsubscript𝛾1𝑛subscript𝑣subscript𝑤𝛾\sup_{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}|{\widehat{A}}_{n}^{(r)}|\ll n^{r+3}\gamma_{1}^{n}|v|_{\infty}|w|_{\gamma} and sup¯|B^n,k(r)|nr+3γ1n|v|γ|w|much-less-thansubscriptsupremum¯superscriptsubscript^𝐵𝑛𝑘𝑟superscript𝑛𝑟3superscriptsubscript𝛾1𝑛subscript𝑣𝛾subscript𝑤\sup_{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}|{\widehat{B}}_{n,k}^{(r)}|\ll n^{r+3}\gamma_{1}^{n}|v|_{\gamma}|w|_{\infty}. Combining these with Corollary 4.21 and substituting into Lemma 4.11, we have shown that ρ^v,w::subscript^𝜌𝑣𝑤\hat{\rho}_{v,w}:{\mathbb{H}}\to{\mathbb{C}} extends to ρ^v,w:¯:subscript^𝜌𝑣𝑤¯\hat{\rho}_{v,w}:{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{C}}. Moreover, we have shown that for every r𝑟r\in{\mathbb{N}} there exists C,α>0𝐶𝛼0C,\,\alpha>0 such that

|ρ^v,w(r)(s)|C(|b|+1)αvγwγfor s=a+ib with a[0,1],superscriptsubscript^𝜌𝑣𝑤𝑟𝑠𝐶superscript𝑏1𝛼subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾for s=a+ib with a[0,1],|\hat{\rho}_{v,w}^{(r)}(s)|\leq C(|b|+1)^{\alpha}\|v\|_{\gamma}\|w\|_{\gamma}\quad\text{for $s=a+ib\in{\mathbb{C}}$ with $a\in[0,1]$,}

for all v,wγ(Yφ)𝑣𝑤subscript𝛾superscript𝑌𝜑v,w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}) with Yφv𝑑μφ=0subscriptsuperscript𝑌𝜑𝑣differential-dsuperscript𝜇𝜑0\int_{Y^{\varphi}}v\,d\mu^{\varphi}=0. The result now follows from Lemma 4.1 and Remark 4.2. ∎

5 Polynomial mixing for skew product Gibbs-Markov flows

In this section, we consider skew product Gibbs-Markov flows Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} for which the roof function φ:Y+:𝜑𝑌superscript\varphi:Y\to{\mathbb{R}}^{+} satisfies μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) for some β>1𝛽1\beta>1. For such flows, we prove Theorem 3.2, namely that absence of approximate eigenfunctions is a sufficient condition to obtain the mixing rate O(t(β1))𝑂superscript𝑡𝛽1O(t^{-(\beta-1)}).

If f::𝑓f:{\mathbb{R}}\to{\mathbb{R}} is integrable, we write f(a(t))𝑓𝑎𝑡f\in{\mathcal{R}}(a(t)) if the inverse Fourier transform of f𝑓f is O(a(t))𝑂𝑎𝑡O(a(t)). We also write (tp)superscript𝑡𝑝{\mathcal{R}}(t^{-p}) instead of ((t+1)p)superscript𝑡1𝑝{\mathcal{R}}((t+1)^{-p}) for p>0𝑝0p>0.

Proposition 5.1 ( [26, Proposition 8.2] )

Let g::𝑔g:{\mathbb{R}}\to{\mathbb{R}} be an integrable function such that g(b)0𝑔𝑏0g(b)\to 0 as b±𝑏plus-or-minusb\to\pm\infty. If |f(q)|gsuperscript𝑓𝑞𝑔|f^{(q)}|\leq g, then f(|g|1tq)𝑓subscript𝑔1superscript𝑡𝑞f\in{\mathcal{R}}(|g|_{1}\,t^{-q}). ∎

The convolution fg𝑓𝑔f\star g of two integrable functions f,g:[0,):𝑓𝑔0f,g:[0,\infty)\to{\mathbb{R}} is defined to be (fg)(t)=0tf(x)g(tx)𝑑x𝑓𝑔𝑡superscriptsubscript0𝑡𝑓𝑥𝑔𝑡𝑥differential-d𝑥(f\star g)(t)=\int_{0}^{t}f(x)g(t-x)\,dx.

Proposition 5.2 ( [26, Proposition 8.4] )

Fix b>a>0𝑏𝑎0b>a>0 with b>1𝑏1b>1. Suppose that f,g:[0,):𝑓𝑔0f,g:[0,\infty)\to{\mathbb{R}} are integrable and there exist constants C,D>0𝐶𝐷0C,D>0 such that |f(t)|C(t+1)a𝑓𝑡𝐶superscript𝑡1𝑎|f(t)|\leq C(t+1)^{-a} and |g(t)|D(t+1)b𝑔𝑡𝐷superscript𝑡1𝑏|g(t)|\leq D(t+1)^{-b} for t0𝑡0t\geq 0. Then there exists a constant K>0𝐾0K>0 depending only on a𝑎a and b𝑏b such that |(fg)(t)|CDK(t+1)a𝑓𝑔𝑡𝐶𝐷𝐾superscript𝑡1𝑎|(f\star g)(t)|\leq CDK(t+1)^{-a} for t0𝑡0t\geq 0. ∎

Proposition 5.3

Define f(b)=b1(eibφ1)𝑓𝑏superscript𝑏1superscript𝑒𝑖𝑏𝜑1f(b)=b^{-1}(e^{-ib\varphi}-1) for b{0}𝑏0b\in{\mathbb{R}}\setminus\{0\}. Then there exists C>0𝐶0C>0 such that 1Y¯kf(q)(b)θCinfY¯kφq+η|b|(1η)subscriptnormsubscript1subscript¯𝑌𝑘superscript𝑓𝑞𝑏𝜃𝐶subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝑞𝜂superscript𝑏1𝜂\|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}f^{(q)}(b)\|_{\theta}\leq C{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{q+\eta}|b|^{-(1-\eta)} for all b{0}𝑏0b\in{\mathbb{R}}\setminus\{0\}.

Proof.

This is contained in the proof of [26, Proposition 8.13]. ∎

5.1 Modified estimate for Rj+1V¯jsuperscript𝑅𝑗1subscript¯𝑉𝑗R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}

Proposition 5.4

There exists C>0𝐶0C>0 such that

Rj+1V¯j(q)(ib)θCγ1j/3vγ|b|(1η),subscriptnormsuperscript𝑅𝑗1superscriptsubscript¯𝑉𝑗𝑞𝑖𝑏𝜃𝐶superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾superscript𝑏1𝜂\textstyle\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}^{(q)}(ib)\|_{\theta}\leq C\gamma_{1}^{j/3}\|v\|_{\gamma}\,|b|^{-(1-\eta)},

for all vγ(Yφ)𝑣subscript𝛾superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma}(Y^{\varphi}) such that v𝑣v is independent of u𝑢u, and all b0𝑏0b\neq 0, j0𝑗0j\geq 0.

Proof.

Recall that

V^j(s)=esφFjΔjvs=0φFjes(φFju)𝑑uΔjv=0φFjesu𝑑uΔjv.subscript^𝑉𝑗𝑠superscript𝑒𝑠𝜑superscript𝐹𝑗subscriptΔ𝑗subscript𝑣𝑠superscriptsubscript0𝜑superscript𝐹𝑗superscript𝑒𝑠𝜑superscript𝐹𝑗𝑢differential-d𝑢subscriptΔ𝑗𝑣superscriptsubscript0𝜑superscript𝐹𝑗superscript𝑒𝑠𝑢differential-d𝑢subscriptΔ𝑗𝑣{\widehat{V}}_{j}(s)=e^{-s\varphi\circ F^{j}}\Delta_{j}v_{s}=\int_{0}^{\varphi\circ F^{j}}e^{-s(\varphi\circ F^{j}-u)}\,du\;\Delta_{j}v=\int_{0}^{\varphi\circ F^{j}}e^{-su}\,du\;\Delta_{j}v.

Hence RjV¯j(s)=0φesu𝑑uRj(Δjv)=s1(esφ1)Rj(Δjv)superscript𝑅𝑗subscript¯𝑉𝑗𝑠superscriptsubscript0𝜑superscript𝑒𝑠𝑢differential-d𝑢superscript𝑅𝑗subscriptΔ𝑗𝑣superscript𝑠1superscript𝑒𝑠𝜑1superscript𝑅𝑗subscriptΔ𝑗𝑣R^{j}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(s)=\int_{0}^{\varphi}e^{-su}\,du\,R^{j}(\Delta_{j}v)=-s^{-1}(e^{-s\varphi}-1)R^{j}(\Delta_{j}v). It follows that

Rj+1V¯j(ib)=iR(f(b)Rj(Δjv)),superscript𝑅𝑗1subscript¯𝑉𝑗𝑖𝑏𝑖𝑅𝑓𝑏superscript𝑅𝑗subscriptΔ𝑗𝑣R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(ib)=iR\big{(}f(b)R^{j}(\Delta_{j}v)\big{)}, (5.1)

where f(b)=b1(eibφ1)𝑓𝑏superscript𝑏1superscript𝑒𝑖𝑏𝜑1f(b)=b^{-1}(e^{-ib\varphi}-1).

Let dY¯𝑑¯𝑌d\in{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} be a j𝑗j-cylinder and let y,yd𝑦superscript𝑦𝑑y,y^{\prime}\in d. Then the arguments in the proof of Proposition 4.12(a,b) show that

|Δjv(y)|γ1jvγφ(Fjy)η,|Δjv(y)Δjv(y)|γ1s(y,y)jvγφ(Fjy)η.formulae-sequencemuch-less-thansubscriptΔ𝑗𝑣𝑦superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂much-less-thansubscriptΔ𝑗𝑣𝑦subscriptΔ𝑗𝑣superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂|\Delta_{j}v(y)|\ll\gamma_{1}^{j}\|v\|_{\gamma}\,\varphi(F^{j}y)^{\eta},\qquad|\Delta_{j}v(y)-\Delta_{j}v(y^{\prime})|\ll\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma}\,\varphi(F^{j}y)^{\eta}.

On the other hand, |Δjv(y)Δjv(y)|γ1jvγφ(Fjy)ηmuch-less-thansubscriptΔ𝑗𝑣𝑦subscriptΔ𝑗𝑣superscript𝑦superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂|\Delta_{j}v(y)-\Delta_{j}v(y^{\prime})|\ll\gamma_{1}^{j}\|v\|_{\gamma}\,\varphi(F^{j}y)^{\eta}, so by (4.2),

|Δjv(y)Δjv(y)|γ1j/3θs(y,y)vγφ(Fjy)η.much-less-thansubscriptΔ𝑗𝑣𝑦subscriptΔ𝑗𝑣superscript𝑦superscriptsubscript𝛾1𝑗3superscript𝜃𝑠𝑦superscript𝑦subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂|\Delta_{j}v(y)-\Delta_{j}v(y^{\prime})|\ll\gamma_{1}^{j/3}\theta^{s(y,y^{\prime})}\|v\|_{\gamma}\,\varphi(F^{j}y)^{\eta}.

Using (3.3), it follows that

|1d(1Y¯kF¯j)Δjv|γ1jvγsupY¯kφη2C1γ1jvγinfY¯kφη,much-less-thansubscriptsubscript1𝑑subscript1subscript¯𝑌𝑘superscript¯𝐹𝑗subscriptΔ𝑗𝑣superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾subscriptsupremumsubscript¯𝑌𝑘superscript𝜑𝜂2subscript𝐶1superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝜂|1_{d}(1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})\Delta_{j}v|_{\infty}\ll\gamma_{1}^{j}\|v\|_{\gamma}\,{\textstyle\sup_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{\eta}\leq 2C_{1}\gamma_{1}^{j}\|v\|_{\gamma}\,{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{\eta},

and similarly,

|1d(1Y¯kF¯j)Δjv|θγ1j/3vγinfY¯kφη,1d(1Y¯kF¯j)Δjvθγ1j/3vγinfY¯kφη.formulae-sequencemuch-less-thansubscriptsubscript1𝑑subscript1subscript¯𝑌𝑘superscript¯𝐹𝑗subscriptΔ𝑗𝑣𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝜂much-less-thansubscriptnormsubscript1𝑑subscript1subscript¯𝑌𝑘superscript¯𝐹𝑗subscriptΔ𝑗𝑣𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝜂|1_{d}(1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})\Delta_{j}v|_{\theta}\ll\gamma_{1}^{j/3}\|v\|_{\gamma}\,{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{\eta},\qquad\|1_{d}(1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})\Delta_{j}v\|_{\theta}\ll\gamma_{1}^{j/3}\|v\|_{\gamma}\,{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{\eta}.

By Proposition 4.4,

1Y¯kRj(Δjv)θ=Rj((1Y¯kF¯j)Δjv)θγ1j/3vγinfY¯kφη.subscriptnormsubscript1subscript¯𝑌𝑘superscript𝑅𝑗subscriptΔ𝑗𝑣𝜃subscriptnormsuperscript𝑅𝑗subscript1subscript¯𝑌𝑘superscript¯𝐹𝑗subscriptΔ𝑗𝑣𝜃much-less-thansuperscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝜂\|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}R^{j}(\Delta_{j}v)\|_{\theta}=\|R^{j}\big{(}(1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})\Delta_{j}v\big{)}\|_{\theta}\ll\gamma_{1}^{j/3}\|v\|_{\gamma}\,{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{\eta}.

Hence by Proposition 5.3,

1Y¯kf(q)(b)Rj(Δjv)θsubscriptnormsubscript1subscript¯𝑌𝑘superscript𝑓𝑞𝑏superscript𝑅𝑗subscriptΔ𝑗𝑣𝜃\displaystyle\|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}f^{(q)}(b)R^{j}(\Delta_{j}v)\|_{\theta} infY¯kφq+η|b|(1η)1Y¯kRj(Δjv)θmuch-less-thanabsentsubscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝑞𝜂superscript𝑏1𝜂subscriptnormsubscript1subscript¯𝑌𝑘superscript𝑅𝑗subscriptΔ𝑗𝑣𝜃\displaystyle\ll{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{q+\eta}|b|^{-(1-\eta)}\|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}R^{j}(\Delta_{j}v)\|_{\theta}
γ1j/3vγinfY¯kφq+2η|b|(1η).much-less-thanabsentsuperscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscriptinfimumsubscript¯𝑌𝑘superscript𝜑𝑞2𝜂superscript𝑏1𝜂\displaystyle\ll\gamma_{1}^{j/3}\|v\|_{\gamma}\,{\textstyle\inf_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}}\varphi^{q+2\eta}|b|^{-(1-\eta)}.

Applying Proposition 4.4 once more and using (5.1),

Rj+1V¯j(q)(ib)θ=R(f(q)(b)Rj(Δjv))θsubscriptnormsuperscript𝑅𝑗1superscriptsubscript¯𝑉𝑗𝑞𝑖𝑏𝜃subscriptnorm𝑅superscript𝑓𝑞𝑏superscript𝑅𝑗subscriptΔ𝑗𝑣𝜃\displaystyle\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}^{(q)}(ib)\|_{\theta}=\textstyle\|R\big{(}f^{(q)}(b)R^{j}(\Delta_{j}v)\big{)}\|_{\theta} kμ¯(Y¯k)1Y¯kf(q)(b)Rj(Δjv)θmuch-less-thanabsentsubscript𝑘¯𝜇subscript¯𝑌𝑘subscriptnormsubscript1subscript¯𝑌𝑘superscript𝑓𝑞𝑏superscript𝑅𝑗subscriptΔ𝑗𝑣𝜃\displaystyle\ll\sum_{k}\bar{\mu}({\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k})\|1_{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{k}}f^{(q)}(b)R^{j}(\Delta_{j}v)\|_{\theta}
γ1j/3vγYφq+2η𝑑μ|b|(1η).much-less-thanabsentsuperscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾subscript𝑌superscript𝜑𝑞2𝜂differential-d𝜇superscript𝑏1𝜂\displaystyle\textstyle\ll\gamma_{1}^{j/3}\|v\|_{\gamma}\int_{Y}\varphi^{q+2\eta}\,d\mu\,|b|^{-(1-\eta)}.

as required. ∎

Let V¯j(t):Y¯:subscript¯𝑉𝑗𝑡¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(t):{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}} denote the inverse Laplace transform associated to V^j(s):Y:subscript^𝑉𝑗𝑠𝑌{\widehat{V}}_{j}(s):Y\to{\mathbb{C}}.

Proposition 5.5

There is a constant C𝐶C such that

Rj+1V¯j(t)θCγ1j/3vγ,η(t+1)q,subscriptnormsuperscript𝑅𝑗1subscript¯𝑉𝑗𝑡𝜃𝐶superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡1𝑞\textstyle\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(t)\|_{\theta}\leq C\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,(t+1)^{-q},

for all vγ,η(Yφ)𝑣subscript𝛾𝜂superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}) with v(y,0)0𝑣𝑦00v(y,0)\equiv 0 and all j0𝑗0j\geq 0, t>0𝑡0t>0.

Proof.

For j0𝑗0j\geq 0,

V^j(s)(y)=0φ(Fjy)es(φ(Fjy)u)Δ~jv(y,u)𝑑u=0φ(Fjy)estΔ~jv(y,φ(Fjy)t)𝑑t,subscript^𝑉𝑗𝑠𝑦superscriptsubscript0𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝜑superscript𝐹𝑗𝑦𝑢subscript~Δ𝑗𝑣𝑦𝑢differential-d𝑢superscriptsubscript0𝜑superscript𝐹𝑗𝑦superscript𝑒𝑠𝑡subscript~Δ𝑗𝑣𝑦𝜑superscript𝐹𝑗𝑦𝑡differential-d𝑡{\widehat{V}}_{j}(s)(y)=\int_{0}^{\varphi(F^{j}y)}e^{-s(\varphi(F^{j}y)-u)}\widetilde{\Delta}_{j}v(y,u)\,du=\int_{0}^{\varphi(F^{j}y)}e^{-st}\widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t)\,dt,

so

Vj(t)(y)=1{φ(Fjy)>t}Δ~jv(y,φ(Fjy)t).subscript𝑉𝑗𝑡𝑦subscript1𝜑superscript𝐹𝑗𝑦𝑡subscript~Δ𝑗𝑣𝑦𝜑superscript𝐹𝑗𝑦𝑡V_{j}(t)(y)=1_{\{\varphi(F^{j}y)>t\}}\widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t).

Recall that c=1/(2C1)superscript𝑐12subscript𝐶1c^{\prime}=1/(2C_{1}). Fix a (j+1)𝑗1(j+1)-cylinder d𝑑d. By Proposition 4.12(a), for yd𝑦𝑑y\in d,

|Vj(t)(y)|subscript𝑉𝑗𝑡𝑦\displaystyle|V_{j}(t)(y)| 2C2γ1j1vγφ(Fjy)η1{|1F¯jdφ|>t}absent2subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾𝜑superscriptsuperscript𝐹𝑗𝑦𝜂subscript1subscriptsubscript1superscript¯𝐹𝑗𝑑𝜑𝑡\displaystyle\leq 2C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\,\varphi(F^{j}y)^{\eta}1_{\{|1_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}\,\varphi|_{\infty}>t\}}
4C1C2γ1j1vγinfF¯jdφη1{infF¯jdφ>ct}.absent4subscript𝐶1subscript𝐶2superscriptsubscript𝛾1𝑗1subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡\displaystyle\leq 4C_{1}C_{2}\gamma_{1}^{j-1}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}. (5.2)

For y,yd𝑦superscript𝑦𝑑y,y^{\prime}\in d,

|φ(Fjy)φ(Fjy)|C1infF¯jdφγs(y,y)j.𝜑superscript𝐹𝑗𝑦𝜑superscript𝐹𝑗superscript𝑦subscript𝐶1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝛾𝑠𝑦superscript𝑦𝑗|\varphi(F^{j}y)-\varphi(F^{j}y^{\prime})|\leq C_{1}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi\,\gamma^{s(y,y^{\prime})-j}.

so by Propositions 4.12(b,c), for t[0,φ(Fjy)][0,φ(Fjy)]𝑡0𝜑superscript𝐹𝑗𝑦0𝜑superscript𝐹𝑗superscript𝑦t\in[0,\varphi(F^{j}y)]\cap[0,\varphi(F^{j}y^{\prime})],

|Δ~jv(y,φ(Fjy)t)\displaystyle|\widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t)- Δ~jv(y,φ(Fjy)t)|\displaystyle\widetilde{\Delta}_{j}v(y^{\prime},\varphi(F^{j}y^{\prime})-t)|
4C2γ1s(y,y)jvγinfF¯jdφη+2|v|,η|φ(Fjy)φ(Fjy)|ηabsent4subscript𝐶2superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂2subscript𝑣𝜂superscript𝜑superscript𝐹𝑗𝑦𝜑superscript𝐹𝑗superscript𝑦𝜂\displaystyle\leq 4C_{2}\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}+2|v|_{\infty,\eta}|\varphi(F^{j}y)-\varphi(F^{j}y^{\prime})|^{\eta}
γ1s(y,y)jvγ,ηinfF¯jdφη.much-less-thanabsentsuperscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾𝜂subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂\displaystyle\ll\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma,\eta}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}. (5.3)

Similarly, for t[φ(Fjy),φ(Fjy)]𝑡𝜑superscript𝐹𝑗superscript𝑦𝜑superscript𝐹𝑗𝑦t\in[\varphi(F^{j}y^{\prime}),\varphi(F^{j}y)],

|Δ~j\displaystyle|\widetilde{\Delta}_{j} v(y,φ(Fjy)t)|=|Δ~jv(y,φ(Fjy)t)Δ~jv(y,0)|2|v|,η|φ(Fjy)t|η\displaystyle v(y,\varphi(F^{j}y)-t)|=|\widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t)-\widetilde{\Delta}_{j}v(y,0)|\leq 2|v|_{\infty,\eta}|\varphi(F^{j}y)-t|^{\eta}
2|v|,η|φ(Fjy)φ(Fjy)|ηγ1s(y,y)j|v|,ηinfF¯jdφη.absent2subscript𝑣𝜂superscript𝜑superscript𝐹𝑗𝑦𝜑superscript𝐹𝑗superscript𝑦𝜂much-less-thansuperscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscript𝑣𝜂subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂\displaystyle\qquad\leq 2|v|_{\infty,\eta}|\varphi(F^{j}y)-\varphi(F^{j}y^{\prime})|^{\eta}\ll\gamma_{1}^{s(y,y^{\prime})-j}|v|_{\infty,\eta}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}. (5.4)

For y,yd𝑦superscript𝑦𝑑y,y^{\prime}\in d with φ(Fjy)φ(Fjy)𝜑superscript𝐹𝑗𝑦𝜑superscript𝐹𝑗superscript𝑦\varphi(F^{j}y)\geq\varphi(F^{j}y^{\prime}),

Vj(t)(y)Vj(t)(y)={Δ~jv(y,φ(Fjy)t)Δ~jv(y,φ(Fjy)t),φ(Fjy)>tΔ~jv(y,φ(Fjy)t),φ(Fjy)>tφ(Fjy)0,φ(Fjy)tsubscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦casessubscript~Δ𝑗𝑣𝑦𝜑superscript𝐹𝑗𝑦𝑡subscript~Δ𝑗𝑣superscript𝑦𝜑superscript𝐹𝑗superscript𝑦𝑡𝜑superscript𝐹𝑗superscript𝑦𝑡otherwisesubscript~Δ𝑗𝑣𝑦𝜑superscript𝐹𝑗𝑦𝑡𝜑superscript𝐹𝑗𝑦𝑡𝜑superscript𝐹𝑗superscript𝑦otherwise0𝜑superscript𝐹𝑗𝑦𝑡otherwise\displaystyle V_{j}(t)(y)-V_{j}(t)(y^{\prime})=\begin{cases}\widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t)-\widetilde{\Delta}_{j}v(y^{\prime},\varphi(F^{j}y^{\prime})-t),\quad\varphi(F^{j}y^{\prime})>t\\ \widetilde{\Delta}_{j}v(y,\varphi(F^{j}y)-t),\qquad\qquad\qquad\qquad\varphi(F^{j}y)>t\geq\varphi(F^{j}y^{\prime})\\ 0,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\varphi(F^{j}y)\leq t\end{cases}

If φ(Fjy)>t𝜑superscript𝐹𝑗superscript𝑦𝑡\varphi(F^{j}y^{\prime})>t, then using (5.1),

|Vj(t)(y)Vj(t)(y)|subscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦\displaystyle|V_{j}(t)(y)-V_{j}(t)(y^{\prime})| γ1s(y,y)jvγ,η1{|1F¯jdφ|>t}infF¯jdφηmuch-less-thanabsentsuperscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾𝜂subscript1subscriptsubscript1superscript¯𝐹𝑗𝑑𝜑𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂\displaystyle\ll\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma,\eta}1_{\{|1_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}\,\varphi|_{\infty}>t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}
γ1s(y,y)jvγ,η1{infF¯jdφ>ct}infF¯jdφη.absentsuperscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂\displaystyle\leq\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma,\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}.

If φ(Fjy)>tφ(Fjy)𝜑superscript𝐹𝑗𝑦𝑡𝜑superscript𝐹𝑗superscript𝑦\varphi(F^{j}y)>t\geq\varphi(F^{j}y^{\prime}), then using (5.1),

|Vj(t)(y)Vj(t)(y)|γ1s(y,y)j|v|,η1{infF¯jdφ>ct}infF¯jdφη.much-less-thansubscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscript𝑣𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂|V_{j}(t)(y)-V_{j}(t)(y^{\prime})|\ll\gamma_{1}^{s(y,y^{\prime})-j}|v|_{\infty,\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\varphi^{\eta}.

Hence in all cases,

|Vj(t)(y)Vj(t)(y)|γ1s(y,y)jvγ,η1{infF¯jdφ>ct}infF¯jdφη.much-less-thansubscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑗subscriptnorm𝑣𝛾𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂|V_{j}(t)(y)-V_{j}(t)(y^{\prime})|\ll\gamma_{1}^{s(y,y^{\prime})-j}\|v\|_{\gamma,\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}.

On the other hand, by (5.1), |Vj(t)(y)Vj(t)(y)|γ1jvγinfF¯jdφη1{infF¯jdφ>ct}much-less-thansubscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦superscriptsubscript𝛾1𝑗subscriptnorm𝑣𝛾subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡|V_{j}(t)(y)-V_{j}(t)(y^{\prime})|\ll\gamma_{1}^{j}\|v\|_{\gamma}\,{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}. Combining these estimates and using (4.2),

|Vj(t)(y)Vj(t)(y)|γ1j/3θs(y,y)vγ,η1{infF¯jdφ>ct}infF¯jdφη.much-less-thansubscript𝑉𝑗𝑡𝑦subscript𝑉𝑗𝑡superscript𝑦superscriptsubscript𝛾1𝑗3superscript𝜃𝑠𝑦superscript𝑦subscriptnorm𝑣𝛾𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂|V_{j}(t)(y)-V_{j}(t)(y^{\prime})|\ll\gamma_{1}^{j/3}\theta^{s(y,y^{\prime})}\|v\|_{\gamma,\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}.

Hence

1dVj(t)θγ1j/3vγ,η1{infF¯jdφ>ct}infF¯jdφη.much-less-thansubscriptnormsubscript1𝑑subscript𝑉𝑗𝑡𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂subscript1subscriptinfimumsuperscript¯𝐹𝑗𝑑𝜑superscript𝑐𝑡subscriptinfimumsuperscript¯𝐹𝑗𝑑superscript𝜑𝜂\|1_{d}V_{j}(t)\|_{\theta}\ll\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}1_{\{{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi>c^{\prime}t\}}{\textstyle\inf_{\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}d}}\,\varphi^{\eta}.

By Proposition 4.4,

Rj+1V¯j(t)θsubscriptnormsuperscript𝑅𝑗1subscript¯𝑉𝑗𝑡𝜃\displaystyle\|R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}(t)\|_{\theta} γ1j/3vγ,ηdμ¯(d)1{infdφF¯j>ct}(infdφF¯j)ηmuch-less-thanabsentsuperscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂subscript𝑑¯𝜇𝑑subscript1subscriptinfimum𝑑𝜑superscript¯𝐹𝑗superscript𝑐𝑡superscriptsubscriptinfimum𝑑𝜑superscript¯𝐹𝑗𝜂\displaystyle\ll\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}{\textstyle\sum_{d}}\,\bar{\mu}(d)1_{\{{\textstyle\inf_{d}}\,\varphi\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}>c^{\prime}t\}}({\textstyle\inf_{d}}\,\varphi\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})^{\eta}
γ1j/3vγ,ηY¯1{φF¯j>ct}(φF¯j)η𝑑μ=γ1j/3vγ,ηY1{φ>ct}φη𝑑μ.absentsuperscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂subscript¯𝑌subscript1𝜑superscript¯𝐹𝑗superscript𝑐𝑡superscript𝜑superscript¯𝐹𝑗𝜂differential-d𝜇superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂subscript𝑌subscript1𝜑superscript𝑐𝑡superscript𝜑𝜂differential-d𝜇\displaystyle\leq\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}1_{\{\varphi\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j}>c^{\prime}t\}}(\varphi\circ\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{j})^{\eta}\,d\mu=\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\int_{Y}1_{\{\varphi>c^{\prime}t\}}\varphi^{\eta}\,d\mu.

Now apply Proposition 3.4(b). ∎

Corollary 5.6

Let κ::𝜅\kappa:{\mathbb{R}}\to{\mathbb{R}} be superscript{\mathbb{C}}^{\infty} with |κ(k)(b)|=O((b2+1)1)superscript𝜅𝑘𝑏𝑂superscriptsuperscript𝑏211|\kappa^{(k)}(b)|=O((b^{2}+1)^{-1}) for all k𝑘k\in{\mathbb{N}}. Then κRj+1V¯jθ(γ1j/3vγ,ηtq)subscriptnorm𝜅superscript𝑅𝑗1subscript¯𝑉𝑗𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞\|\kappa R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}\|_{\theta}\in{\mathcal{R}}(\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}) for all vγ,η(Yφ)𝑣subscript𝛾𝜂superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}), j0𝑗0j\geq 0.

Proof.

Write v(y,u)=v0(y)+v1(y,u)𝑣𝑦𝑢subscript𝑣0𝑦subscript𝑣1𝑦𝑢v(y,u)=v_{0}(y)+v_{1}(y,u) where v0(y)=v(y,0)subscript𝑣0𝑦𝑣𝑦0v_{0}(y)=v(y,0). We have the corresponding decomposition V¯j=V¯j,0+V¯j,1subscript¯𝑉𝑗subscript¯𝑉𝑗0subscript¯𝑉𝑗1{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}={\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j,0}+{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j,1}. The function g(b)=κ(b)|b|(1η)𝑔𝑏𝜅𝑏superscript𝑏1𝜂g(b)=\kappa(b)|b|^{-(1-\eta)} is integrable and (κRj+1V¯j,0)(q)(ib)θγ1j/3vγ,ηg(b)much-less-thansubscriptnormsuperscript𝜅superscript𝑅𝑗1subscript¯𝑉𝑗0𝑞𝑖𝑏𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂𝑔𝑏\|(\kappa R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j,0})^{(q)}(ib)\|_{\theta}\ll\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,g(b) by Proposition 5.4, so κRj+1V¯j,0θ(γ1j/3vγtq)subscriptnorm𝜅superscript𝑅𝑗1subscript¯𝑉𝑗0𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾superscript𝑡𝑞\|\kappa R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j,0}\|_{\theta}\in{\mathcal{R}}(\gamma_{1}^{j/3}\|v\|_{\gamma}\,t^{-q}) by Proposition 5.1. Also, κ(tq)𝜅superscript𝑡𝑞\kappa\in{\mathcal{R}}(t^{-q}) by Proposition 5.1, so κRj+1V¯j,1θ(γ1j/3vγ,ηtq)subscriptnorm𝜅superscript𝑅𝑗1subscript¯𝑉𝑗1𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞\|\kappa R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j,1}\|_{\theta}\in{\mathcal{R}}(\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}) by Propositions 5.2 and 5.5. ∎

5.2 Truncation

We proceed in a manner analogous to [26, Section 8.4], replacing φ𝜑\varphi by a bounded roof function. Given N1𝑁1N\geq 1, let Y(N)=j1:infYjφNYj𝑌𝑁subscript:𝑗1subscriptinfimumsubscript𝑌𝑗𝜑𝑁subscript𝑌𝑗Y(N)=\bigcup_{j\geq 1:\inf_{Y_{j}}\varphi\geq N}Y_{j}. Define φ(N)=N𝜑𝑁𝑁\varphi(N)=N on Y(N)𝑌𝑁Y(N) and φ(N)=φ𝜑𝑁𝜑\varphi(N)=\varphi elsewhere. (Unlike [26], it is not sufficient to take φ(N)=min{φ,N}𝜑𝑁𝜑𝑁\varphi(N)=\min\{\varphi,N\}.) Note that φ(N)2C1N𝜑𝑁2subscript𝐶1𝑁\varphi(N)\leq 2C_{1}N by (3.3).

Consider the suspension semiflows Ftsubscript𝐹𝑡F_{t} and FN,tsubscript𝐹𝑁𝑡F_{N,t} on Yφsuperscript𝑌𝜑Y^{\varphi} and Yφ(N)superscript𝑌𝜑𝑁Y^{\varphi(N)} respectively. (Here, FN,tsubscript𝐹𝑁𝑡F_{N,t} is computed modulo the identification (y,φ(N)(y))(Fy,0)similar-to𝑦𝜑𝑁𝑦𝐹𝑦0(y,\varphi(N)(y))\sim(Fy,0) on Yφ(N)superscript𝑌𝜑𝑁Y^{\varphi(N)}.) Let ρv,wsubscript𝜌𝑣𝑤\rho_{v,w} and ρv,wtruncsubscriptsuperscript𝜌trunc𝑣𝑤\rho^{\rm trunc}_{v,w} denote the respective correlation functions. In particular, ρv,wtrunc(t)=Yφ(N)vwFN,t𝑑μφ(N)Yφ(N)v𝑑μφ(N)Yφ(N)w𝑑μφ(N)subscriptsuperscript𝜌trunc𝑣𝑤𝑡subscriptsuperscript𝑌𝜑𝑁𝑣𝑤subscript𝐹𝑁𝑡differential-dsuperscript𝜇𝜑𝑁subscriptsuperscript𝑌𝜑𝑁𝑣differential-dsuperscript𝜇𝜑𝑁subscriptsuperscript𝑌𝜑𝑁𝑤differential-dsuperscript𝜇𝜑𝑁\rho^{\rm trunc}_{v,w}(t)=\int_{Y^{\varphi(N)}}v\,w\circ F_{N,t}\,d\mu^{\varphi(N)}-\int_{Y^{\varphi(N)}}v\,\,d\mu^{\varphi(N)}\int_{Y^{\varphi(N)}}w\,\,d\mu^{\varphi(N)} where the observables v,w:Yφ(N):𝑣𝑤superscript𝑌𝜑𝑁v,w:Y^{\varphi(N)}\to{\mathbb{R}} are the restrictions of v,w:Yφ:𝑣𝑤superscript𝑌𝜑v,w:Y^{\varphi}\to{\mathbb{R}} to Yφ(N)superscript𝑌𝜑𝑁Y^{\varphi(N)}.

Proposition 5.7 ( [26, Proposition 8.19] )

There are constants C,t0>0𝐶subscript𝑡00C,\,t_{0}>0, N01subscript𝑁01N_{0}\geq 1 such that

|ρv,w(t)ρv,wtrunc(t)|C|v||w|(tNβ+N(β1)),subscript𝜌𝑣𝑤𝑡subscriptsuperscript𝜌trunc𝑣𝑤𝑡𝐶subscript𝑣subscript𝑤𝑡superscript𝑁𝛽superscript𝑁𝛽1|\rho_{v,w}(t)-\rho^{\rm trunc}_{v,w}(t)|\leq C|v|_{\infty}|w|_{\infty}(tN^{-\beta}+N^{-(\beta-1)}),

for all v,wL(Yφ)𝑣𝑤superscript𝐿superscript𝑌𝜑v,w\in L^{\infty}(Y^{\varphi}), NN0𝑁subscript𝑁0N\geq N_{0}, t>t0𝑡subscript𝑡0t>t_{0}. ∎

We make the following abuse of notation regarding norms of observables v:Yφ(N):𝑣superscript𝑌𝜑𝑁v:Y^{\varphi(N)}\to{\mathbb{R}}. Define vγ,η=vγ,ηsubscriptnorm𝑣𝛾𝜂subscriptnormsuperscript𝑣𝛾𝜂\|v\|_{\gamma,\eta}=\|v^{\prime}\|_{\gamma,\eta} where vsuperscript𝑣v^{\prime} is the extension of v𝑣v by zero to Yφsuperscript𝑌𝜑Y^{\varphi}. (In other words, the factor of φ𝜑\varphi on the denominator in the definition of |v|γsubscript𝑣𝛾|v|_{\gamma} is not replaced by φ(N)𝜑𝑁\varphi(N).)

With this convention, vγ,η(Yφ)𝑣subscript𝛾𝜂superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}) restricts to v|Yφ(N)γ,η(Yφ(N))evaluated-at𝑣superscript𝑌𝜑𝑁subscript𝛾𝜂superscript𝑌𝜑𝑁v|_{Y^{\varphi(N)}}\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi(N)}) with v|Yφ(N)γ,ηvγ,ηevaluated-atsubscriptdelimited-‖|𝑣superscript𝑌𝜑𝑁𝛾𝜂subscriptnorm𝑣𝛾𝜂\|v|_{Y^{\varphi(N)}}\|_{\gamma,\eta}\leq\|v\|_{\gamma,\eta}. The similar convention applies to observables wγ(Yφ(N))𝑤subscript𝛾superscript𝑌𝜑𝑁w\in{\mathcal{H}}_{\gamma}(Y^{\varphi(N)}). However, restricting wγ,0,m(Yφ)𝑤subscript𝛾0𝑚superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi}) to Yφ(N)superscript𝑌𝜑𝑁Y^{\varphi(N)} need not preserve smoothness in the flow direction. Below we prove:

Lemma 5.8

Assume absence of approximate eigenfunctions. In particular, there is a finite union ZY¯𝑍¯𝑌Z\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} of partition elements such that the corresponding finite subsystem Z0subscript𝑍0Z_{0} does not support approximate eigenfunctions. Choose N1|1Zφ|+3subscript𝑁1subscriptsubscript1𝑍𝜑3N_{1}\geq|1_{Z}\varphi|_{\infty}+3.

There exist m1𝑚1m\geq 1, C>0𝐶0C>0 such that

|ρv,wtrunc(t)|Cvγ,ηwγ,0,mt(β1),subscriptsuperscript𝜌trunc𝑣𝑤𝑡𝐶subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾0𝑚superscript𝑡𝛽1|\rho^{\rm trunc}_{v,w}(t)|\leq C\|v\|_{\gamma,\eta}\|w\|_{\gamma,0,m}\,t^{-(\beta-1)},

for all vγ,η(Yφ(N))𝑣subscript𝛾𝜂superscript𝑌𝜑𝑁v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi(N)}), wγ,0,m(Yφ(N))𝑤subscript𝛾0𝑚superscript𝑌𝜑𝑁w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi(N)}), NN1𝑁subscript𝑁1N\geq N_{1}, t>1𝑡1t>1.

Proof of Theorem 3.2   Let m1𝑚1m\geq 1, N13subscript𝑁13N_{1}\geq 3 be as in Lemma 5.8. As discussed above, the observable v:Yφ:𝑣superscript𝑌𝜑v:Y^{\varphi}\to{\mathbb{C}} restricts to an observable v:Yφ(N):𝑣superscript𝑌𝜑𝑁v:Y^{\varphi(N)}\to{\mathbb{C}} with no increase in the value of vγ,ηsubscriptnorm𝑣𝛾𝜂\|v\|_{\gamma,\eta}, but restricting wγ,0,m(Yφ)𝑤subscript𝛾0𝑚superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi}) to Yφ(N)superscript𝑌𝜑𝑁Y^{\varphi(N)} need not preserve smoothness in the flow direction. To circumvent this, following [25, 26] we define an approximating observable wN:Yφ(N):subscript𝑤𝑁superscript𝑌𝜑𝑁w_{N}:Y^{\varphi(N)}\to{\mathbb{R}}, NN1𝑁subscript𝑁1N\geq N_{1},

wN(y,u)={w(y,u)(y,u)Y(N)×[N2,N]j=02m+1(uN+2)jdN,j(y)(y,u)Y(N)×[N2,N1]w(y,u+φ(y)N)(y,u)Y(N)×(N1,N],subscript𝑤𝑁𝑦𝑢cases𝑤𝑦𝑢𝑦𝑢𝑌𝑁𝑁2𝑁superscriptsubscript𝑗02𝑚1superscript𝑢𝑁2𝑗subscript𝑑𝑁𝑗𝑦𝑦𝑢𝑌𝑁𝑁2𝑁1𝑤𝑦𝑢𝜑𝑦𝑁𝑦𝑢𝑌𝑁𝑁1𝑁w_{N}(y,u)=\begin{cases}w(y,u)&(y,u)\not\in Y(N)\times[N-2,N]\\ \sum_{j=0}^{2m+1}(u-N+2)^{j}d_{N,j}(y)&(y,u)\in Y(N)\times[N-2,N-1]\\ w(y,u+\varphi(y)-N)&(y,u)\in Y(N)\times(N-1,N]\end{cases},

where the dN,j(y)subscript𝑑𝑁𝑗𝑦d_{N,j}(y) are linear combinations of tjw(y,N2)superscriptsubscript𝑡𝑗𝑤𝑦𝑁2\partial_{t}^{j}w(y,N-2) and tjw(y,φ(y)1)superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦1\partial_{t}^{j}w(y,\varphi(y)-1), j=0,,m𝑗0𝑚j=0,\dots,m, with coefficients independent of y𝑦y and N𝑁N uniquely specified by the requirements tjwN(y,N2)=tjw(y,N2)superscriptsubscript𝑡𝑗subscript𝑤𝑁𝑦𝑁2superscriptsubscript𝑡𝑗𝑤𝑦𝑁2\partial_{t}^{j}w_{N}(y,N-2)=\partial_{t}^{j}w(y,N-2) and tjwN(y,N1)=tjw(y,φ(y)1)superscriptsubscript𝑡𝑗subscript𝑤𝑁𝑦𝑁1superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦1\partial_{t}^{j}w_{N}(y,N-1)=\partial_{t}^{j}w(y,\varphi(y)-1) for j=0,,m𝑗0𝑚j=0,\dots,m. 333In fact dN,j(y)=(1/j!)tjw(y,N2)subscript𝑑𝑁𝑗𝑦1𝑗superscriptsubscript𝑡𝑗𝑤𝑦𝑁2d_{N,j}(y)=(1/j!)\partial_{t}^{j}w(y,N-2) for 0jm0𝑗𝑚0\leq j\leq m but the remaining formulas are messier. When m=1𝑚1m=1, for instance, dN,2(y)=3w(y,N2)2tw(y,N2)+3w(y,φ(y)1)tw(y,φ(y)1)subscript𝑑𝑁2𝑦3𝑤𝑦𝑁22subscript𝑡𝑤𝑦𝑁23𝑤𝑦𝜑𝑦1subscript𝑡𝑤𝑦𝜑𝑦1d_{N,2}(y)=-3w(y,N-2)-2\partial_{t}w(y,N-2)+3w(y,\varphi(y)-1)-\partial_{t}w(y,\varphi(y)-1), dN,3(y)=2w(y,N2)+tw(y,N2)2w(y,φ(y)1)+tw(y,φ(y)1)subscript𝑑𝑁3𝑦2𝑤𝑦𝑁2subscript𝑡𝑤𝑦𝑁22𝑤𝑦𝜑𝑦1subscript𝑡𝑤𝑦𝜑𝑦1d_{N,3}(y)=2w(y,N-2)+\partial_{t}w(y,N-2)-2w(y,\varphi(y)-1)+\partial_{t}w(y,\varphi(y)-1).

It is immediate from the definitions that wNsubscript𝑤𝑁w_{N} is m𝑚m-times differentiable in the flow direction. We claim that wNγ,0,mCwγ,0,m+1subscriptnormsubscript𝑤𝑁𝛾0𝑚superscript𝐶subscriptnorm𝑤𝛾0𝑚1\|w_{N}\|_{\gamma,0,m}\leq C^{\prime}\|w\|_{\gamma,0,m+1} for some constant Csuperscript𝐶C^{\prime} independent of N𝑁N. By Lemma 5.8,

|ρv,wNtrunc(t)|CCvγ,ηwγ,0,m+1t(β1).subscriptsuperscript𝜌trunc𝑣subscript𝑤𝑁𝑡𝐶superscript𝐶subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾0𝑚1superscript𝑡𝛽1|\rho^{\rm trunc}_{v,w_{N}}(t)|\leq CC^{\prime}\|v\|_{\gamma,\eta}\|w\|_{\gamma,0,m+1}\,t^{-(\beta-1)}.

Also,

|ρv,wtrunc\displaystyle|\rho^{\rm trunc}_{v,w} (t)ρv,wNtrunc(t)||v|(|w|+|wN|)μφ(N)(FN,t1SN)\displaystyle(t)-\rho^{\rm trunc}_{v,w_{N}}(t)|\leq|v|_{\infty}(|w|_{\infty}+|w_{N}|_{\infty})\mu^{\varphi(N)}(F_{N,t}^{-1}S_{N})
=|v|(|w|+|wN|)μφ(N)(SN)2|v||w|μ(φ>N)|v||w|Nβ,absentsubscript𝑣subscript𝑤subscriptsubscript𝑤𝑁superscript𝜇𝜑𝑁subscript𝑆𝑁2subscript𝑣subscript𝑤𝜇𝜑𝑁much-less-thansubscript𝑣subscript𝑤superscript𝑁𝛽\displaystyle=|v|_{\infty}(|w|_{\infty}+|w_{N}|_{\infty})\mu^{\varphi(N)}(S_{N})\leq 2|v|_{\infty}|w|_{\infty}\mu(\varphi>N)\ll|v|_{\infty}|w|_{\infty}\,N^{-\beta},

so

|ρv,wtrunc(t)|vγ,ηwγ,0,m+1(t(β1)+Nβ).much-less-thansubscriptsuperscript𝜌trunc𝑣𝑤𝑡subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾0𝑚1superscript𝑡𝛽1superscript𝑁𝛽|\rho^{\rm trunc}_{v,w}(t)|\ll\|v\|_{\gamma,\eta}\|w\|_{\gamma,0,m+1}\,(t^{-(\beta-1)}+N^{-\beta}).

Taking N=[t]𝑁delimited-[]𝑡N=[t], the result follows directly from Proposition 5.7.

It remains to verify the claim. Fix k{0,,m}𝑘0𝑚k\in\{0,\dots,m\}. Let (y,u),(y,u)Y(N)×[N2,N1]𝑦𝑢superscript𝑦𝑢𝑌𝑁𝑁2𝑁1(y,u),\,(y^{\prime},u)\in Y(N)\times[N-2,N-1], where y,y𝑦superscript𝑦y,y^{\prime} lie in the same partition element. Then

|tkwN(y,u)|superscriptsubscript𝑡𝑘subscript𝑤𝑁𝑦𝑢\displaystyle|\partial_{t}^{k}w_{N}(y,u)| (2m+1)!j=02m+1|dN,j(y)|absent2𝑚1superscriptsubscript𝑗02𝑚1subscript𝑑𝑁𝑗𝑦\displaystyle\textstyle\leq(2m+1)!\sum_{j=0}^{2m+1}|d_{N,j}(y)|
Cj=0m(|tjw(y,N2)|+|tjw(y,φ(y)1)|)2Cwγ,0,m,absent𝐶superscriptsubscript𝑗0𝑚superscriptsubscript𝑡𝑗𝑤𝑦𝑁2superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦12𝐶subscriptnorm𝑤𝛾0𝑚\displaystyle\leq\textstyle C\sum_{j=0}^{m}(|\partial_{t}^{j}w(y,N-2)|+|\partial_{t}^{j}w(y,\varphi(y)-1)|)\leq 2C\|w\|_{\gamma,0,m},

where C𝐶C is a constant independent of N𝑁N. Also, by (3.3), for 0jm0𝑗𝑚0\leq j\leq m

|tjw(y,φ(y)1)tjw(y,φ(y)1)||tj+1w||φ(y)φ(y)|C1|tj+1w|φ(y)γs(y,y).superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦1superscriptsubscript𝑡𝑗𝑤𝑦𝜑superscript𝑦1subscriptsuperscriptsubscript𝑡𝑗1𝑤𝜑𝑦𝜑superscript𝑦subscript𝐶1subscriptsuperscriptsubscript𝑡𝑗1𝑤𝜑𝑦superscript𝛾𝑠𝑦superscript𝑦\displaystyle|\partial_{t}^{j}w(y,\varphi(y)-1)-\partial_{t}^{j}w(y,\varphi(y^{\prime})-1)|\leq|\partial_{t}^{j+1}w|_{\infty}|\varphi(y)-\varphi(y^{\prime})|\leq C_{1}|\partial_{t}^{j+1}w|_{\infty}\varphi(y)\gamma^{s(y,y^{\prime})}.

Hence

|tk\displaystyle|\partial_{t}^{k} wN(y,u)|tkwN(y,u)|(2m+1)!j=02m+1|dN,j(y)dN,j(y)|subscript𝑤𝑁𝑦𝑢superscriptsubscript𝑡𝑘subscript𝑤𝑁superscript𝑦𝑢2𝑚1superscriptsubscript𝑗02𝑚1subscript𝑑𝑁𝑗𝑦subscript𝑑𝑁𝑗superscript𝑦\displaystyle w_{N}(y,u)|-\partial_{t}^{k}w_{N}(y^{\prime},u)|\leq\textstyle(2m+1)!\sum_{j=0}^{2m+1}|d_{N,j}(y)-d_{N,j}(y^{\prime})|
Cj=0m(|tjw(y,N2)tjw(y,N2)|+|tjw(y,φ(y)1)tjw(y,φ(y)1)|)absent𝐶superscriptsubscript𝑗0𝑚superscriptsubscript𝑡𝑗𝑤𝑦𝑁2superscriptsubscript𝑡𝑗𝑤superscript𝑦𝑁2superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦1superscriptsubscript𝑡𝑗𝑤superscript𝑦𝜑superscript𝑦1\displaystyle\leq\textstyle C\sum_{j=0}^{m}\big{(}|\partial_{t}^{j}w(y,N-2)-\partial_{t}^{j}w(y^{\prime},N-2)|+|\partial_{t}^{j}w(y,\varphi(y)-1)-\partial_{t}^{j}w(y^{\prime},\varphi(y^{\prime})-1)|\big{)}
2Cj=0m|tjw|γφ(y){d(y,y)+γs(y,y)}+Cj=0m|tjw(y,φ(y)1)tjw(y,φ(y)1)|absent2𝐶superscriptsubscript𝑗0𝑚subscriptsuperscriptsubscript𝑡𝑗𝑤𝛾𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝐶superscriptsubscript𝑗0𝑚superscriptsubscript𝑡𝑗𝑤𝑦𝜑𝑦1superscriptsubscript𝑡𝑗𝑤𝑦𝜑superscript𝑦1\displaystyle\textstyle\leq 2C\sum_{j=0}^{m}|\partial_{t}^{j}w|_{\gamma}\varphi(y)\{d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}\}+C\sum_{j=0}^{m}|\partial_{t}^{j}w(y,\varphi(y)-1)-\partial_{t}^{j}w(y,\varphi(y^{\prime})-1)|
2Cw|γ,0,mφ(y){d(y,y)+γs(y,y)}+CC1wγ,0,m+1φ(y)γs(y,y)absent2𝐶subscriptdelimited-‖|𝑤𝛾0𝑚𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝐶subscript𝐶1subscriptnorm𝑤𝛾0𝑚1𝜑𝑦superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq 2C\|w|_{\gamma,0,m}\varphi(y)\{d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}\}+CC_{1}\|w\|_{\gamma,0,m+1}\varphi(y)\gamma^{s(y,y^{\prime})}
3CC1w|γ,0,m+1φ(y){d(y,y)+γs(y,y)}.absent3𝐶subscript𝐶1subscriptdelimited-‖|𝑤𝛾0𝑚1𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq 3CC_{1}\|w|_{\gamma,0,m+1}\varphi(y)\{d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}\}.

This completes the verification of the claim on the region Y(N)×[N2,N1]𝑌𝑁𝑁2𝑁1Y(N)\times[N-2,N-1] and the other regions are easier to treat. ∎

Our strategy for proving Lemma 5.8 is identical to that for [26, Lemma 8.20]. The first step is to show that the inverse Laplace transform of ρv,wtrunc^^subscriptsuperscript𝜌trunc𝑣𝑤\widehat{\rho^{\rm trunc}_{v,w}} can be computed using the imaginary axis as the contour of integration.

Proposition 5.9

Let NN1𝑁subscript𝑁1N\geq N_{1}, v,wγ(Yφ(N))𝑣𝑤subscript𝛾superscript𝑌𝜑𝑁v,w\in{\mathcal{H}}_{\gamma}(Y^{\varphi(N)}). Then there exists ϵ>0italic-ϵ0\epsilon>0, C>0𝐶0C>0, α0𝛼0\alpha\geq 0, such that ρv,wtrunc^^subscriptsuperscript𝜌trunc𝑣𝑤\widehat{\rho^{\rm trunc}_{v,w}} is continuous on {Res[0,ϵ]}Re𝑠0italic-ϵ\{\operatorname{Re}s\in[0,\epsilon]\} and |ρv,wtrunc^(s)|C(|b|+1)α^subscriptsuperscript𝜌trunc𝑣𝑤𝑠𝐶superscript𝑏1𝛼|\widehat{\rho^{\rm trunc}_{v,w}}(s)|\leq C(|b|+1)^{\alpha} for all s=a+ib𝑠𝑎𝑖𝑏s=a+ib with a[0,ϵ]𝑎0italic-ϵa\in[0,\epsilon].

Proof.

In this proof, the constant C𝐶C is not required to be uniform in N𝑁N. Consequently, the estimates are very straightforward compared to other estimates in this section.

The desired properties for ρv,wtrunc^^subscriptsuperscript𝜌trunc𝑣𝑤\widehat{\rho^{\rm trunc}_{v,w}} will hold provided they are verified for all the constituent parts in Lemma 4.11. Note that if f𝑓f is integrable on [0,)0[0,\infty), then f^^𝑓\hat{f} satisfies the required properties with α=0𝛼0\alpha=0. Hence the estimate in Proposition 4.16 already suffices for W¯ksubscript¯𝑊𝑘{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}. Also, the proof of Proposition 4.19 suffices after truncation since φr+3superscript𝜑𝑟3\varphi^{r+3} becomes (2C1N)r+2φsuperscript2subscript𝐶1𝑁𝑟2𝜑(2C_{1}N)^{r+2}\varphi. (Actually, the factor φr+3superscript𝜑𝑟3\varphi^{r+3} is easily improved to φr+1+2ηsuperscript𝜑𝑟12𝜂\varphi^{r+1+2\eta} which is integrable when r=0𝑟0r=0 so truncation is not absolutely necessary for the term Rj+1V¯jsuperscript𝑅𝑗1subscript¯𝑉𝑗R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}.)

By definition of N1subscript𝑁1N_{1}, the truncated roof function φ(N)𝜑𝑁\varphi(N) coincides with φ𝜑\varphi on the subsystem Z0subscript𝑍0Z_{0}, so absence of approximate eigenfunctions passes over to the truncated flow for each NN1𝑁subscript𝑁1N\geq N_{1}. Since φ(N)2C1N𝜑𝑁2subscript𝐶1𝑁\varphi(N)\leq 2C_{1}N, all estimates related to R^^𝑅{\widehat{R}} and T^^𝑇{\widehat{T}} in Section 4 now hold for q𝑞q arbitrarily large. Hence the arguments in Section 4 yield the desired properties for 0j,k<Cj,ksubscriptformulae-sequence0𝑗𝑘subscript𝐶𝑗𝑘\sum_{0\leq j,k<\infty}C_{j,k}. Also, it is immediate from the proof of [26, Proposition 6.3] that |J0(t)|Nμ(φ(N)>t)much-less-thansubscript𝐽0𝑡𝑁𝜇𝜑𝑁𝑡|J_{0}(t)|\ll N\mu(\varphi(N)>t) so J0(t)=0subscript𝐽0𝑡0J_{0}(t)=0 for t>N𝑡𝑁t>N and hence J0subscript𝐽0J_{0} is integrable.

It remains to consider the terms Ansubscript𝐴𝑛A_{n} and Bnsubscript𝐵𝑛B_{n}. Here, we must take into account that the factor of φ𝜑\varphi in the definition of γ\|\;\|_{\gamma} is not truncated. Starting from the end of the proof of Proposition 4.14, we obtain

|An(t)|4C1C2Nγ1n1|v||w|γYφηFn1{φn+1>t}𝑑μ.subscript𝐴𝑛𝑡4subscript𝐶1subscript𝐶2𝑁superscriptsubscript𝛾1𝑛1subscript𝑣subscript𝑤𝛾subscript𝑌superscript𝜑𝜂superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡differential-d𝜇\textstyle|A_{n}(t)|\leq 4C_{1}C_{2}N\gamma_{1}^{n-1}|v|_{\infty}|w|_{\gamma}\int_{Y}\varphi^{\eta}\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu.

A simplified version of Proposition 4.13 combined with Proposition 3.4(b) yields

YφηFn1{φn+1>t}𝑑μsubscript𝑌superscript𝜑𝜂superscript𝐹𝑛subscript1subscript𝜑𝑛1𝑡differential-d𝜇\displaystyle\int_{Y}\varphi^{\eta}\circ F^{n}1_{\{\varphi_{n+1}>t\}}\,d\mu j=0nYφηRnj1{φ>t/n}𝑑μabsentsuperscriptsubscript𝑗0𝑛subscript𝑌superscript𝜑𝜂superscript𝑅𝑛𝑗subscript1𝜑𝑡𝑛differential-d𝜇\displaystyle\leq\sum_{j=0}^{n}\int_{Y}\varphi^{\eta}R^{n-j}1_{\{\varphi>t/n\}}\,d\mu
j=0n1|φ|1|R1{φ>t/n}|+Yφη1{φ>t/n}𝑑μnβ+1t(βη).absentsuperscriptsubscript𝑗0𝑛1subscript𝜑1subscript𝑅subscript1𝜑𝑡𝑛subscript𝑌superscript𝜑𝜂subscript1𝜑𝑡𝑛differential-d𝜇much-less-thansuperscript𝑛𝛽1superscript𝑡𝛽𝜂\displaystyle\leq\sum_{j=0}^{n-1}|\varphi|_{1}|R1_{\{\varphi>t/n\}}|_{\infty}+\int_{Y}\varphi^{\eta}1_{\{\varphi>t/n\}}\,d\mu\ll n^{\beta+1}t^{-(\beta-\eta)}.

Hence |An(t)|nβ+1γ1n|v||w|γt(βη)much-less-thansubscript𝐴𝑛𝑡superscript𝑛𝛽1superscriptsubscript𝛾1𝑛subscript𝑣subscript𝑤𝛾superscript𝑡𝛽𝜂|A_{n}(t)|\ll n^{\beta+1}\gamma_{1}^{n}|v|_{\infty}|w|_{\gamma}\,t^{-(\beta-\eta)}. Similarly |Bn,k(t)|nβ+1γ1n|v|γ|w|t(βη)much-less-thansubscript𝐵𝑛𝑘𝑡superscript𝑛𝛽1superscriptsubscript𝛾1𝑛subscript𝑣𝛾subscript𝑤superscript𝑡𝛽𝜂|B_{n,k}(t)|\ll n^{\beta+1}\gamma_{1}^{n}|v|_{\gamma}|w|_{\infty}\,t^{-(\beta-\eta)}. Hence n1Ansubscript𝑛1subscript𝐴𝑛\sum_{n\geq 1}A_{n} and 0k<n<Bn,ksubscript0𝑘𝑛subscript𝐵𝑛𝑘\sum_{0\leq k<n<\infty}B_{n,k} are integrable, completing the proof. ∎

Choose ψ:[0,1]:𝜓01\psi:{\mathbb{R}}\to[0,1] to be Csuperscript𝐶C^{\infty} and compactly supported such that ψ1𝜓1\psi\equiv 1 on a neighbourhood of zero. Let κm(b)=(1ψ(b))(ib)msubscript𝜅𝑚𝑏1𝜓𝑏superscript𝑖𝑏𝑚\kappa_{m}(b)=(1-\psi(b))(ib)^{-m}, m2𝑚2m\geq 2.

Corollary 5.10

Let NN1𝑁subscript𝑁1N\geq N_{1}, mα+2𝑚𝛼2m\geq\alpha+2, vγ(Yφ(N))𝑣subscript𝛾superscript𝑌𝜑𝑁v\in{\mathcal{H}}_{\gamma}(Y^{\varphi(N)}), wγ,0,m(Yφ(N))𝑤subscript𝛾0𝑚superscript𝑌𝜑𝑁w\in{\mathcal{H}}_{\gamma,0,m}(Y^{\varphi(N)}). Then

ρv,wtrunc(t)=ψ(b)eibtρv,wtrunc^(ib)𝑑b+κm(b)eibtρv,tmwtrunc^(ib)𝑑b.subscriptsuperscript𝜌trunc𝑣𝑤𝑡superscriptsubscript𝜓𝑏superscript𝑒𝑖𝑏𝑡^subscriptsuperscript𝜌trunc𝑣𝑤𝑖𝑏differential-d𝑏superscriptsubscriptsubscript𝜅𝑚𝑏superscript𝑒𝑖𝑏𝑡^subscriptsuperscript𝜌trunc𝑣superscriptsubscript𝑡𝑚𝑤𝑖𝑏differential-d𝑏\rho^{\rm trunc}_{v,w}(t)=\int_{-\infty}^{\infty}\psi(b)e^{ibt}\widehat{\rho^{\rm trunc}_{v,w}}(ib)db+\int_{-\infty}^{\infty}\kappa_{m}(b)e^{ibt}\widehat{\rho^{\rm trunc}_{v,\partial_{t}^{m}w}}(ib)db.
Proof.

As in [26, Section 6.1], we can suppose without loss that ρv,wtruncsuperscriptsubscript𝜌𝑣𝑤trunc\rho_{v,w}^{\rm trunc} vanishes for t𝑡t near zero, so that

ρv,wtrunc^(s)=smρv,tmwtrunc^(s)for all s.^subscriptsuperscript𝜌trunc𝑣𝑤𝑠superscript𝑠𝑚^subscriptsuperscript𝜌trunc𝑣superscriptsubscript𝑡𝑚𝑤𝑠for all s\widehat{\rho^{\rm trunc}_{v,w}}(s)=s^{-m}\widehat{\rho^{\rm trunc}_{v,\partial_{t}^{m}w}}(s)\quad\text{for all $s\in{\mathbb{H}}$}. (5.5)

By Proposition 5.9, it follows as in the proof of [26, Lemma 6.2] that

ρv,wtrunc(t)=eibtρv,wtrunc^(ib)𝑑b=ψ(b)eibtρv,wtrunc^(ib)𝑑b+(1ψ(b))eibtρv,wtrunc^(ib)𝑑b.subscriptsuperscript𝜌trunc𝑣𝑤𝑡superscriptsubscriptsuperscript𝑒𝑖𝑏𝑡^subscriptsuperscript𝜌trunc𝑣𝑤𝑖𝑏differential-d𝑏superscriptsubscript𝜓𝑏superscript𝑒𝑖𝑏𝑡^subscriptsuperscript𝜌trunc𝑣𝑤𝑖𝑏differential-d𝑏superscriptsubscript1𝜓𝑏superscript𝑒𝑖𝑏𝑡^subscriptsuperscript𝜌trunc𝑣𝑤𝑖𝑏differential-d𝑏\textstyle\rho^{\rm trunc}_{v,w}(t)=\int_{-\infty}^{\infty}e^{ibt}\widehat{\rho^{\rm trunc}_{v,w}}(ib)db=\int_{-\infty}^{\infty}\psi(b)e^{ibt}\widehat{\rho^{\rm trunc}_{v,w}}(ib)db+\int_{-\infty}^{\infty}(1-\psi(b))e^{ibt}\widehat{\rho^{\rm trunc}_{v,w}}(ib)db.

By Proposition 5.9, equation (5.5) extends to ¯{0}¯0{\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{H}}\mkern-1.5mu}\mkern 1.5mu}\setminus\{0\} and the result follows. ∎

From now on we suppress the superscript “trunctrunc{\rm trunc}” for sake of readability. Notation R^^𝑅{\widehat{R}}, T^^𝑇{\widehat{T}} and so on refers to the operators obtained using φ(N)𝜑𝑁\varphi(N) instead of φ𝜑\varphi. We end this subsection by recalling some further estimates from [26]. The first is a uniform version of Proposition 4.8.

Proposition 5.11 ( [26, Proposition 8.27] )

Assume absence of approximate eigenfunctions. Then there exists m2𝑚2m\geq 2 such that

κm(b)T^(ib)θ(tq)uniformly in NN1.subscriptnormsubscript𝜅𝑚𝑏^𝑇𝑖𝑏𝜃superscript𝑡𝑞uniformly in NN1.\|\kappa_{m}(b){\widehat{T}}(ib)\|_{\theta}\in{\mathcal{R}}(t^{-q})\quad\text{uniformly in $N\geq N_{1}$.}

The remaining estimates in this subsection are required when b𝑏b is close to zero. By Proposition 4.9, for each N1𝑁1N\geq 1 there exists δ>0𝛿0\delta>0 such that

R^(ib)=λ(b)P(b)+R^(ib)Q(b)for |b|<δ,^𝑅𝑖𝑏𝜆𝑏𝑃𝑏^𝑅𝑖𝑏𝑄𝑏for |b|<δ,{\widehat{R}}(ib)=\lambda(b)P(b)+{\widehat{R}}(ib)Q(b)\quad\text{for $|b|<\delta$,}

where λ,P𝜆𝑃\lambda,\,P and Q=IP𝑄𝐼𝑃Q=I-P are Csuperscript𝐶C^{\infty} on (δ,δ)𝛿𝛿(-\delta,\delta) and λ(0)=1𝜆01\lambda(0)=1, λ(0)=i|φ(N)|1superscript𝜆0𝑖subscript𝜑𝑁1\lambda^{\prime}(0)=-i|\varphi(N)|_{1} and P(0)v=Y¯v𝑑μ¯𝑃0𝑣subscript¯𝑌𝑣differential-d¯𝜇P(0)v=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}v\,d\bar{\mu}. In fact, as shown in [26, Section 8.5], δ>0𝛿0\delta>0 can be chosen uniformly in N𝑁N. Moreover, R^(q)(ib)θsubscriptnormsuperscript^𝑅𝑞𝑖𝑏𝜃\|{\widehat{R}}^{(q)}(ib)\|_{\theta} is bounded uniformly in N𝑁N on (δ,δ)𝛿𝛿(-\delta,\delta), so λ,P,Q𝜆𝑃𝑄\lambda,\,P,\,Q are Cqsuperscript𝐶𝑞C^{q} uniformly in N𝑁N on (δ,δ)𝛿𝛿(-\delta,\delta).

Define

P~(b)=b1(P(b)P(0)),λ~=b1(1λ(b)).formulae-sequence~𝑃𝑏superscript𝑏1𝑃𝑏𝑃0~𝜆superscript𝑏11𝜆𝑏\widetilde{P}(b)=b^{-1}(P(b)-P(0)),\qquad\tilde{\lambda}=b^{-1}(1-\lambda(b)).
Proposition 5.12

There exists a constant C>0𝐶0C>0, uniform in N1𝑁1N\geq 1, such that

(λ~1)(q)(ib)θ,(λ~1P~)(q)(ib)θC|b|(1η)for |b|<δ,formulae-sequencesubscriptnormsuperscriptsuperscript~𝜆1𝑞𝑖𝑏𝜃subscriptnormsuperscriptsuperscript~𝜆1~𝑃𝑞𝑖𝑏𝜃𝐶superscript𝑏1𝜂for |b|<δ,\|(\tilde{\lambda}^{-1})^{(q)}(ib)\|_{\theta},\;\|(\tilde{\lambda}^{-1}\widetilde{P})^{(q)}(ib)\|_{\theta}\leq C|b|^{-(1-\eta)}\quad\text{for $|b|<\delta$,}
Proof.

By [26, Proposition 8.18], P~(q1)(b)θ{|b|(1η)q1<β2η1q1<β1much-less-thansubscriptnormsuperscript~𝑃subscript𝑞1𝑏𝜃casessuperscript𝑏1𝜂subscript𝑞1𝛽2𝜂1subscript𝑞1𝛽1\|\widetilde{P}^{(q_{1})}(b)\|_{\theta}\ll\begin{cases}|b|^{-(1-\eta)}&q_{1}<\beta-2\eta\\ 1&q_{1}<\beta-1\end{cases}. The argument in the proof of [26, Proposition 8.26] gives the same estimates for λ~1superscript~𝜆1\tilde{\lambda}^{-1} completing the estimates for λ~1P~superscript~𝜆1~𝑃\tilde{\lambda}^{-1}\widetilde{P}. ∎

5.3 Proof of Lemma 5.8

Let ψ𝜓\psi and κmsubscript𝜅𝑚\kappa_{m} be as in Corollary 5.10 with the extra property that suppψ(δ,δ)supp𝜓𝛿𝛿\operatorname{supp}\psi\subset(-\delta,\delta). By Proposition 5.1,

ψ,κm(tp)for all p>0m2.formulae-sequence𝜓subscript𝜅𝑚superscript𝑡𝑝for all p>0m2\psi,\,\kappa_{m}\in{\mathcal{R}}(t^{-p})\quad\text{for all $p>0$, $m\geq 2$}. (5.6)

By Corollary 5.10, we need to show that ψ(b)ρ^v,w(ib)(vγ,ηwγt(β1))𝜓𝑏subscript^𝜌𝑣𝑤𝑖𝑏subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝛽1\psi(b)\hat{\rho}_{v,w}(ib)\in{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-(\beta-1)}) and κm(b)ρ^v,w(ib)(vγ,ηwγt(β1))subscript𝜅𝑚𝑏subscript^𝜌𝑣𝑤𝑖𝑏subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝛽1\kappa_{m}(b)\hat{\rho}_{v,w}(ib)\in{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-(\beta-1)}) for all vγ,η(Yφ(N))𝑣subscript𝛾𝜂superscript𝑌𝜑𝑁v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi(N)}), wγ(Yφ(N))𝑤subscript𝛾superscript𝑌𝜑𝑁w\in{\mathcal{H}}_{\gamma}(Y^{\varphi(N)}), uniformly in NN1𝑁subscript𝑁1N\geq N_{1}.

Let A^=n=1A^n^𝐴superscriptsubscript𝑛1subscript^𝐴𝑛{\widehat{A}}=\sum_{n=1}^{\infty}{\widehat{A}}_{n}, B^=n=1k=0n1B^n,k^𝐵superscriptsubscript𝑛1superscriptsubscript𝑘0𝑛1subscript^𝐵𝑛𝑘{\widehat{B}}=\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}{\widehat{B}}_{n,k}, C^=j,k=0C^j,k^𝐶superscriptsubscript𝑗𝑘0subscript^𝐶𝑗𝑘{\widehat{C}}=\sum_{j,k=0}^{\infty}{\widehat{C}}_{j,k}. By Lemma 4.11, it remains to show that each of the terms

ψJ^,ψA^,ψB^,ψC^;κmJ^,κmA^,κmB^,κmC^,𝜓^𝐽𝜓^𝐴𝜓^𝐵𝜓^𝐶subscript𝜅𝑚^𝐽subscript𝜅𝑚^𝐴subscript𝜅𝑚^𝐵subscript𝜅𝑚^𝐶\displaystyle\psi{\widehat{J}},\quad\psi{\widehat{A}},\quad\psi{\widehat{B}},\quad\psi{\widehat{C}};\qquad\kappa_{m}{\widehat{J}},\quad\kappa_{m}{\widehat{A}},\quad\kappa_{m}{\widehat{B}},\quad\kappa_{m}{\widehat{C}},

lies in (vγ,ηwγt(β1))subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝛽1{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-(\beta-1)}) uniformly in N1𝑁1N\geq 1.

By Propositions 4.34.14 and 4.15, J^,A^,B^(vγwγt(β1))^𝐽^𝐴^𝐵subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾superscript𝑡𝛽1{\widehat{J}},\,{\widehat{A}},\,{\widehat{B}}\in{\mathcal{R}}(\|v\|_{\gamma}\|w\|_{\gamma}\,t^{-(\beta-1)}). (Estimates such as these that hold even before truncation are clearly independent of N𝑁N.) By (5.6) and Proposition 5.2, uniformly in N1𝑁1N\geq 1,

ψJ^,ψA^,ψB^,κmJ^,κmA^,κmB^(vγwγt(β1)).𝜓^𝐽𝜓^𝐴𝜓^𝐵subscript𝜅𝑚^𝐽subscript𝜅𝑚^𝐴subscript𝜅𝑚^𝐵subscriptnorm𝑣𝛾subscriptnorm𝑤𝛾superscript𝑡𝛽1\psi{\widehat{J}},\;\psi{\widehat{A}},\;\psi{\widehat{B}},\;\kappa_{m}{\widehat{J}},\;\kappa_{m}{\widehat{A}},\;\kappa_{m}{\widehat{B}}\in{\mathcal{R}}(\|v\|_{\gamma}\|w\|_{\gamma}\,t^{-(\beta-1)}).

Hence it remains to estimate ψC^𝜓^𝐶\psi{\widehat{C}} and κmC^subscript𝜅𝑚^𝐶\kappa_{m}{\widehat{C}}. The next lemma provides the desired estimates and completes the proof of Lemma 5.8 (recall that q>β1𝑞𝛽1q>\beta-1).

Lemma 5.13

Assume absence of approximate eigenfunctions. There exists N11subscript𝑁11N_{1}\geq 1, m2𝑚2m\geq 2, such that after truncation, uniformly in NN1𝑁subscript𝑁1N\geq N_{1},

  • (a)

    κmC^(vγ,ηwγtq)subscript𝜅𝑚^𝐶subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝑞\kappa_{m}{\widehat{C}}\in{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-q}), and

  • (b)

    ψC^(vγ,ηwγt(β1))𝜓^𝐶subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝛽1\psi{\widehat{C}}\in{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-(\beta-1)}),

for all t>1𝑡1t>1, vγ,η(Yφ)𝑣subscript𝛾𝜂superscript𝑌𝜑v\in{\mathcal{H}}_{\gamma,\eta}(Y^{\varphi}), wγ(Yφ)𝑤subscript𝛾superscript𝑌𝜑w\in{\mathcal{H}}_{\gamma}(Y^{\varphi}).

Proof.

(a) Let =max{jk1,0}𝑗𝑘10\ell=\max\{j-k-1,0\} and recall that

C^j,k=Y¯Dj,W¯k𝑑μ¯,Dj,=R^T^Rj+1V¯j.formulae-sequencesubscript^𝐶𝑗𝑘subscript¯𝑌subscript𝐷𝑗subscript¯𝑊𝑘differential-d¯𝜇subscript𝐷𝑗superscript^𝑅^𝑇superscript𝑅𝑗1subscript¯𝑉𝑗\textstyle{\widehat{C}}_{j,k}=\int_{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}D_{j,\ell}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\bar{\mu},\qquad D_{j,\ell}={\widehat{R}}^{\ell}{\widehat{T}}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}.

By Proposition 5.11, we can choose m2𝑚2m\geq 2 such that κm5T^θ(tq)subscriptnormsubscript𝜅𝑚5^𝑇𝜃superscript𝑡𝑞\|\kappa_{m-5}{\widehat{T}}\|_{\theta}\in{\mathcal{R}}(t^{-q}) uniformly in NN1𝑁subscript𝑁1N\geq N_{1}. Write κm=κ3κm5κ2subscript𝜅𝑚subscript𝜅3subscript𝜅𝑚5subscript𝜅2\kappa_{m}=\kappa_{3}\kappa_{m-5}\kappa_{2}, where κisubscript𝜅𝑖\kappa_{i} is Csuperscript𝐶C^{\infty}, vanishes in a neighborhood of zero, and is O(|b|i)𝑂superscript𝑏𝑖O(|b|^{-i}). Then

|κmDj,|κ3R^θκm5T^θκ2Rj+1V¯jθ.subscriptsubscript𝜅𝑚subscript𝐷𝑗subscriptnormsubscript𝜅3superscript^𝑅𝜃subscriptnormsubscript𝜅𝑚5^𝑇𝜃subscriptnormsubscript𝜅2superscript𝑅𝑗1subscript¯𝑉𝑗𝜃|\kappa_{m}D_{j,\ell}|_{\infty}\leq\|\kappa_{3}{\widehat{R}}^{\ell}\|_{\theta}\|\kappa_{m-5}{\widehat{T}}\|_{\theta}\|\kappa_{2}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}\|_{\theta}.

The estimates for R^superscript^𝑅{\widehat{R}}^{\ell} and Rj+1V¯jsuperscript𝑅𝑗1subscript¯𝑉𝑗R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j} in Proposition 4.17 and Corollary 5.6 hold even before truncation and hence are uniform in N1𝑁1N\geq 1. Using (5.6) and Propositions 5.1 and 5.2,

κ3R^θ((+1)βtq),κ2Rj+1V¯jθ(γ1j/3vγ,ηtq),formulae-sequencesubscriptnormsubscript𝜅3superscript^𝑅𝜃superscript1𝛽superscript𝑡𝑞subscriptnormsubscript𝜅2superscript𝑅𝑗1subscript¯𝑉𝑗𝜃superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞\|\kappa_{3}{\widehat{R}}^{\ell}\|_{\theta}\in{\mathcal{R}}((\ell+1)^{\beta}t^{-q}),\quad\|\kappa_{2}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}\|_{\theta}\in{\mathcal{R}}\big{(}\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}\big{)},

uniformly in N1𝑁1N\geq 1. Since q>1𝑞1q>1, it follows from Proposition 5.2 that uniformly in NN1𝑁subscript𝑁1N\geq N_{1},

|κmDj,|subscriptsubscript𝜅𝑚subscript𝐷𝑗\displaystyle|\kappa_{m}D_{j,\ell}|_{\infty} ((+1)βtqtqγ1j/3vγ,ηtq)((+1)βγ1j/3vγ,ηtq).absentsuperscript1𝛽superscript𝑡𝑞superscript𝑡𝑞superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞superscript1𝛽superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞\displaystyle\in{\mathcal{R}}\big{(}(\ell+1)^{\beta}t^{-q}\star t^{-q}\star\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}\big{)}\in{\mathcal{R}}\big{(}(\ell+1)^{\beta}\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}\big{)}.

Also, |W¯k|1((k+1)β+1γ1kwγtq)subscriptsubscript¯𝑊𝑘1superscript𝑘1𝛽1superscriptsubscript𝛾1𝑘subscriptnorm𝑤𝛾superscript𝑡𝑞|{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}|_{1}\in{\mathcal{R}}\big{(}(k+1)^{\beta+1}\gamma_{1}^{k}\|w\|_{\gamma}\,t^{-q}\big{)} by Proposition 4.16 and this is uniform in N1𝑁1N\geq 1. Applying Proposition 5.2 once more, uniformly in NN1𝑁subscript𝑁1N\geq N_{1},

κmC^j,k((j+1)βγ1j/3(k+1)β+1γ1kvγ,ηwγtq),subscript𝜅𝑚subscript^𝐶𝑗𝑘superscript𝑗1𝛽superscriptsubscript𝛾1𝑗3superscript𝑘1𝛽1superscriptsubscript𝛾1𝑘subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝑞\kappa_{m}{\widehat{C}}_{j,k}\in{\mathcal{R}}((j+1)^{\beta}\gamma_{1}^{j/3}(k+1)^{\beta+1}\gamma_{1}^{k}\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-q}),

and part (a) follows.
(b) As in the proof of Proposition 4.20, we write

Dj,=(1λ)1YV^j𝑑μ+Qj,,subscript𝐷𝑗superscript1𝜆1subscript𝑌subscript^𝑉𝑗differential-d𝜇subscript𝑄𝑗\textstyle D_{j,\ell}=(1-\lambda)^{-1}\int_{Y}{\widehat{V}}_{j}\,d\mu+Q_{j,\ell},

where

Qj,=((λ1++1)P(0)+λQ2+R^Q1)Rj+1V¯j.subscript𝑄𝑗superscript𝜆11𝑃0superscript𝜆subscript𝑄2superscript^𝑅subscript𝑄1superscript𝑅𝑗1subscript¯𝑉𝑗Q_{j,\ell}=\big{(}-(\lambda^{\ell-1}+\dots+1)P(0)+\lambda^{\ell}Q_{2}+{\widehat{R}}^{\ell}Q_{1}\big{)}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}.

Here, Q2=(1λ)1(PP(0))=λ~1P~subscript𝑄2superscript1𝜆1𝑃𝑃0superscript~𝜆1~𝑃Q_{2}=(1-\lambda)^{-1}(P-P(0))=\tilde{\lambda}^{-1}\widetilde{P}.

By Proposition 4.18,

C^=j,kYDj,W¯k𝑑μ=j,kYQj,W¯k𝑑μ+(1λ)1I0kYW¯k𝑑μ,^𝐶subscript𝑗𝑘subscript𝑌subscript𝐷𝑗subscript¯𝑊𝑘differential-d𝜇subscript𝑗𝑘subscript𝑌subscript𝑄𝑗subscript¯𝑊𝑘differential-d𝜇superscript1𝜆1subscript𝐼0subscript𝑘subscript𝑌subscript¯𝑊𝑘differential-d𝜇\textstyle{\widehat{C}}=\sum_{j,k}\int_{Y}D_{j,\ell}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu=\sum_{j,k}\int_{Y}Q_{j,\ell}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu+(1-\lambda)^{-1}I_{0}\sum_{k}\int_{Y}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu,

where I0(s)=Y0φ(y)es(φ(y)u)v(y,u)𝑑u𝑑μsubscript𝐼0𝑠subscript𝑌superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝜑𝑦𝑢𝑣𝑦𝑢differential-d𝑢differential-d𝜇I_{0}(s)=\int_{Y}\int_{0}^{\varphi(y)}e^{-s(\varphi(y)-u)}v(y,u)\,du\,d\mu.

Choose ψ1subscript𝜓1\psi_{1} to be Csuperscript𝐶C^{\infty} with compact support such that ψ11subscript𝜓11\psi_{1}\equiv 1 on suppψsupp𝜓\operatorname{supp}\psi. By Corollary 5.6 and Propositions 4.17 and 5.12, uniformly in N1𝑁1N\geq 1,

|ψλQ2Rj+1V¯j||ψλ|ψ1λ~1P~θψ1Rj+1V¯jθ((+1)βγ1j/3vγ,ηtq).subscript𝜓superscript𝜆subscript𝑄2superscript𝑅𝑗1subscript¯𝑉𝑗𝜓superscript𝜆subscriptnormsubscript𝜓1superscript~𝜆1~𝑃𝜃subscriptnormsubscript𝜓1superscript𝑅𝑗1subscript¯𝑉𝑗𝜃superscript1𝛽superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞|\psi\lambda^{\ell}Q_{2}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}|_{\infty}\leq|\psi\lambda^{\ell}|\|\psi_{1}\tilde{\lambda}^{-1}\widetilde{P}\|_{\theta}\|\psi_{1}R^{j+1}{\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu}\!_{j}\|_{\theta}\in{\mathcal{R}}\big{(}(\ell+1)^{\beta}\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}\big{)}.

The other terms in Qj,subscript𝑄𝑗Q_{j,\ell} are simpler and we obtain that |ψQj,|((+1)βγ1j/3vγ,ηtq)subscript𝜓subscript𝑄𝑗superscript1𝛽superscriptsubscript𝛾1𝑗3subscriptnorm𝑣𝛾𝜂superscript𝑡𝑞|\psi Q_{j,\ell}|_{\infty}\in{\mathcal{R}}\big{(}(\ell+1)^{\beta}\gamma_{1}^{j/3}\|v\|_{\gamma,\eta}\,t^{-q}\big{)}. Hence by Proposition 4.16, uniformly in N1𝑁1N\geq 1,

ψj,kYQj,W¯k𝑑μ(vγ,ηwγtq),kYW¯k𝑑μ(wγtq).formulae-sequence𝜓subscript𝑗𝑘subscript𝑌subscript𝑄𝑗subscript¯𝑊𝑘differential-d𝜇subscriptnorm𝑣𝛾𝜂subscriptnorm𝑤𝛾superscript𝑡𝑞subscript𝑘subscript𝑌subscript¯𝑊𝑘differential-d𝜇subscriptnorm𝑤𝛾superscript𝑡𝑞\psi\sum_{j,k}\int_{Y}Q_{j,\ell}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu\in{\mathcal{R}}(\|v\|_{\gamma,\eta}\|w\|_{\gamma}\,t^{-q}),\qquad\sum_{k}\int_{Y}{\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu}\!_{k}\,d\mu\in{\mathcal{R}}\big{(}\|w\|_{\gamma}\,t^{-q}\big{)}.

To complete the proof, it remains to estimate ψ(1λ)1I0𝜓superscript1𝜆1subscript𝐼0\psi(1-\lambda)^{-1}I_{0}. Recall from (4.4) that I0(0)=0subscript𝐼000I_{0}(0)=0, so (1λ)1I0=λ~1I^1superscript1𝜆1subscript𝐼0superscript~𝜆1subscript^𝐼1(1-\lambda)^{-1}I_{0}=\tilde{\lambda}^{-1}\widehat{I}_{1} where

I^1(s)=s1(I0(s)I0(0))=s1Y0φ(y)(es(φ(y)u)1)v(y,u)𝑑u𝑑μ,subscript^𝐼1𝑠superscript𝑠1subscript𝐼0𝑠subscript𝐼00superscript𝑠1subscript𝑌superscriptsubscript0𝜑𝑦superscript𝑒𝑠𝜑𝑦𝑢1𝑣𝑦𝑢differential-d𝑢differential-d𝜇\widehat{I}_{1}(s)=s^{-1}(I_{0}(s)-I_{0}(0))=s^{-1}\int_{Y}\int_{0}^{\varphi(y)}(e^{-s(\varphi(y)-u)}-1)v(y,u)\,du\,d\mu,

with inverse Laplace transform I1(t)=Y0φ(y)1{φ(y)>t+u}v(y,u)𝑑u𝑑μsubscript𝐼1𝑡subscript𝑌superscriptsubscript0𝜑𝑦subscript1𝜑𝑦𝑡𝑢𝑣𝑦𝑢differential-d𝑢differential-d𝜇I_{1}(t)=-\int_{Y}\int_{0}^{\varphi(y)}1_{\{\varphi(y)>t+u\}}v(y,u)\,du\,d\mu. By Proposition 3.4(b), |I1(t)||v|Yφ1{φ>t}𝑑μ|v|t(β1)subscript𝐼1𝑡subscript𝑣subscript𝑌𝜑subscript1𝜑𝑡differential-d𝜇much-less-thansubscript𝑣superscript𝑡𝛽1|I_{1}(t)|\leq|v|_{\infty}\int_{Y}\varphi 1_{\{\varphi>t\}}\,d\mu\ll|v|_{\infty}\,t^{-(\beta-1)}, uniformly in N1𝑁1N\geq 1, so I^1(|v|t(β1))subscript^𝐼1subscript𝑣superscript𝑡𝛽1\widehat{I}_{1}\in{\mathcal{R}}(|v|_{\infty}\,t^{-(\beta-1)}). Combining this with Proposition 5.12, we obtain that

ψ(1λ)1I0=ψλ~1I^1(tq|v|t(β1))(|v|t(β1)),𝜓superscript1𝜆1subscript𝐼0𝜓superscript~𝜆1subscript^𝐼1superscript𝑡𝑞subscript𝑣superscript𝑡𝛽1subscript𝑣superscript𝑡𝛽1\psi(1-\lambda)^{-1}I_{0}=\psi\tilde{\lambda}^{-1}\widehat{I}_{1}\in{\mathcal{R}}(t^{-q}\star|v|_{\infty}\,t^{-(\beta-1)})\in{\mathcal{R}}(|v|_{\infty}\,t^{-(\beta-1)}),

uniformly in N1𝑁1N\geq 1. ∎

6 General Gibbs-Markov flows

In this section, we assume the setup from Section 3 but we drop the requirement that φ𝜑\varphi is constant along stable leaves.

In Subsection 6.1, we introduce a criterion, condition (H), that enables us to reduce to the skew product Gibbs-Markov maps studied in Sections 34 and 5. This leads to an enlarged class of Gibbs-Markov flows for which we can prove results on mixing rates (Theorem 6.4 below). In Subsection 6.2, we recall criteria for absence of approximate eigenfunctions based on periodic data.

6.1 Condition (H)

Let F:YY:𝐹𝑌𝑌F:Y\to Y be a map as in Section 3 with quotient Gibbs-Markov map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}, and define Y~j=YjY~subscript~𝑌𝑗subscript𝑌𝑗~𝑌{\widetilde{Y}}\!_{j}=Y_{j}\cap{\widetilde{Y}}. Let φ:Y+:𝜑𝑌superscript\varphi:Y\to{\mathbb{R}}^{+} be an integrable roof function with infφ>1infimum𝜑1\inf\varphi>1 and associated suspension flow Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi}.

We no longer assume that φ𝜑\varphi is constant along stable leaves. Instead of condition (3.3) we require that

|φ(y)φ(y)|C1infYjφγs(y,y)for all y,yY~jj1.𝜑𝑦𝜑superscript𝑦subscript𝐶1subscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠𝑦superscript𝑦for all y,yY~jj1.|\varphi(y)-\varphi(y^{\prime})|\leq C_{1}{\textstyle\inf_{Y_{j}}}\varphi\,\gamma^{s(y,y^{\prime})}\quad\text{for all $y,y^{\prime}\in{\widetilde{Y}}\!_{j}$, $j\geq 1$.} (6.1)

(Clearly, if φ𝜑\varphi is constant along stable leaves, then conditions (3.3) and (6.1) are identical.)

Recall that π:YY~:𝜋𝑌~𝑌\pi:Y\to{\widetilde{Y}} is the projection along stable leaves. Define

χ(y)=n=0(φ(Fnπy)φ(Fny)),𝜒𝑦superscriptsubscript𝑛0𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦\textstyle\chi(y)=\sum_{n=0}^{\infty}(\varphi(F^{n}\pi y)-\varphi(F^{n}y)),

for all yY𝑦𝑌y\in Y such that the series converges absolutely. We assume

  • (H)
    • (a)

      The series converges almost surely on Y𝑌Y and χL(Y)𝜒superscript𝐿𝑌\chi\in L^{\infty}(Y).

    • (b)

      There are constants C31subscript𝐶31C_{3}\geq 1, γ(0,1)𝛾01\gamma\in(0,1) such that

      |χ(y)χ(y)|C3(d(y,y)+γs(y,y)) for all y,yY.𝜒𝑦𝜒superscript𝑦subscript𝐶3𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦 for all y,yY.|\chi(y)-\chi(y^{\prime})|\leq C_{3}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})})\quad\text{ for all $y,y^{\prime}\in Y$.}

When conditions (6.1) and (H) are satisfied, we call Ftsubscript𝐹𝑡F_{t} a Gibbs-Markov flow. (If φ𝜑\varphi is constant along stable leaves then χ=0𝜒0\chi=0, so every skew product Gibbs-Markov flow is a Gibbs-Markov flow.)

Since infφ>0infimum𝜑0\inf\varphi>0, it follows that φn=j=0n1φFj4|χ|+1subscript𝜑𝑛superscriptsubscript𝑗0𝑛1𝜑superscript𝐹𝑗4subscript𝜒1\varphi_{n}=\sum_{j=0}^{n-1}\varphi\circ F^{j}\geq 4|\chi|_{\infty}+1 for all n𝑛n sufficiently large. For simplicity we suppose from now on that infφ4|χ|+1infimum𝜑4subscript𝜒1\inf\varphi\geq 4|\chi|_{\infty}+1 (otherwise, replace F𝐹F by Fnsuperscript𝐹𝑛F^{n}).

Define

φ~=φ+χχF.~𝜑𝜑𝜒𝜒𝐹\tilde{\varphi}=\varphi+\chi-\chi\circ F. (6.2)

Note that infφ~infφ2|χ|1infimum~𝜑infimum𝜑2subscript𝜒1\inf\tilde{\varphi}\geq\inf\varphi-2|\chi|_{\infty}\geq 1 and Yφ~𝑑μ=Yφ𝑑μsubscript𝑌~𝜑differential-d𝜇subscript𝑌𝜑differential-d𝜇\int_{Y}\tilde{\varphi}\,d\mu=\int_{Y}\varphi\,d\mu, so φ~:Y+:~𝜑𝑌superscript\tilde{\varphi}:Y\to{\mathbb{R}}^{+} is an integrable roof function. Hence we can define the suspension flow F~t:Yφ~Yφ~:subscript~𝐹𝑡superscript𝑌~𝜑superscript𝑌~𝜑\widetilde{F}_{t}:Y^{\tilde{\varphi}}\to Y^{\tilde{\varphi}}. Also, a calculation shows that φ~(y)=n=0(φ(Fnπy)φ(FnπFy))~𝜑𝑦superscriptsubscript𝑛0𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝜋𝐹𝑦\tilde{\varphi}(y)=\sum_{n=0}^{\infty}(\varphi(F^{n}\pi y)-\varphi(F^{n}\pi{Fy})), so φ~~𝜑\tilde{\varphi} is constant along stable leaves and we can define the quotient roof function φ¯:Y¯+:¯𝜑¯𝑌superscript{\bar{\varphi}}:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mathbb{R}}^{+} with quotient semiflow F¯t:Y¯φ~Y¯φ~:subscript¯𝐹𝑡superscript¯𝑌~𝜑superscript¯𝑌~𝜑\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t}:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\tilde{\varphi}}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}^{\tilde{\varphi}}.

In the remainder of this section, we prove that F~tsubscript~𝐹𝑡\widetilde{F}_{t} is a skew product Gibbs-Markov flow (and hence F¯tsubscript¯𝐹𝑡\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t} is a Gibbs-Markov semiflow), and show that (super)polynomial decay of correlations for F~tsubscript~𝐹𝑡\widetilde{F}_{t} is inherited by Ftsubscript𝐹𝑡F_{t}.

Proposition 6.1

Let Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} be a Gibbs-Markov flow. Then F~t:Yφ~Yφ~:subscript~𝐹𝑡superscript𝑌~𝜑superscript𝑌~𝜑\widetilde{F}_{t}:Y^{\tilde{\varphi}}\to Y^{\tilde{\varphi}} is a skew product Gibbs-Markov flow.

Proof.

We verify that the setup in Section 3 holds. All the conditions on the map F:YY:𝐹𝑌𝑌F:Y\to Y are satisfied by assumption. Hence it suffices to check that φ~~𝜑\tilde{\varphi} satisfies condition (3.3).

Let y,yY~j𝑦superscript𝑦subscript~𝑌𝑗y,y^{\prime}\in{\widetilde{Y}}\!_{j} for some j1𝑗1j\geq 1. By (3.2), d(y,y)C2γs(y,y)𝑑𝑦superscript𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦d(y,y^{\prime})\leq C_{2}\gamma^{s(y,y^{\prime})} and d(Fy,Fy)C2γs(y,y)1𝑑𝐹𝑦𝐹superscript𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦1d(Fy,Fy^{\prime})\leq C_{2}\gamma^{s(y,y^{\prime})-1}. By (H)(b), |χ(y)χ(y)|2C2C3γs(y,y)𝜒𝑦𝜒superscript𝑦2subscript𝐶2subscript𝐶3superscript𝛾𝑠𝑦superscript𝑦|\chi(y)-\chi(y^{\prime})|\leq 2C_{2}C_{3}\gamma^{s(y,y^{\prime})} and |χ(Fy)χ(Fy)|2C2C3γs(y,y)1𝜒𝐹𝑦𝜒𝐹superscript𝑦2subscript𝐶2subscript𝐶3superscript𝛾𝑠𝑦superscript𝑦1|\chi(Fy)-\chi(Fy^{\prime})|\leq 2C_{2}C_{3}\gamma^{s(y,y^{\prime})-1}. Hence by (6.1) and (6.2),

|φ~(y)φ~(y)||φ(y)φ(y)|+|χ(y)χ(y)|+|χ(Fy)χ(Fy)|infYjφγs(y,y).~𝜑𝑦~𝜑superscript𝑦𝜑𝑦𝜑superscript𝑦𝜒𝑦𝜒superscript𝑦𝜒𝐹𝑦𝜒𝐹superscript𝑦much-less-thansubscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠𝑦superscript𝑦|\tilde{\varphi}(y)-\tilde{\varphi}(y^{\prime})|\leq|\varphi(y)-\varphi(y^{\prime})|+|\chi(y)-\chi(y^{\prime})|+|\chi(Fy)-\chi(Fy^{\prime})|\ll{\textstyle\inf_{Y_{j}}}\varphi\,\gamma^{s(y,y^{\prime})}.

Also, infYjφinfYjφ~+2|χ|infYjφ~+12infφinfYjφ~+12infYjφsubscriptinfimumsubscript𝑌𝑗𝜑subscriptinfimumsubscript𝑌𝑗~𝜑2subscript𝜒subscriptinfimumsubscript𝑌𝑗~𝜑12infimum𝜑subscriptinfimumsubscript𝑌𝑗~𝜑12subscriptinfimumsubscript𝑌𝑗𝜑{\textstyle\inf_{Y_{j}}}\varphi\leq{\textstyle\inf_{Y_{j}}}\tilde{\varphi}+2|\chi|_{\infty}\leq{\textstyle\inf_{Y_{j}}}\tilde{\varphi}+\frac{1}{2}\inf\varphi\leq{\textstyle\inf_{Y_{j}}}\tilde{\varphi}+\frac{1}{2}{\textstyle\inf_{Y_{j}}}\varphi. Hence infYjφ2infYjφ~subscriptinfimumsubscript𝑌𝑗𝜑2subscriptinfimumsubscript𝑌𝑗~𝜑{\textstyle\inf_{Y_{j}}}\varphi\leq 2{\textstyle\inf_{Y_{j}}}\tilde{\varphi} and |φ~(y)φ~(y)|infYjφ~γs(y,y)much-less-than~𝜑𝑦~𝜑superscript𝑦subscriptinfimumsubscript𝑌𝑗~𝜑superscript𝛾𝑠𝑦superscript𝑦|\tilde{\varphi}(y)-\tilde{\varphi}(y^{\prime})|\ll{\textstyle\inf_{Y_{j}}}\tilde{\varphi}\,\gamma^{s(y,y^{\prime})} as required. ∎

Corollary 6.2

There is a constant C>0𝐶0C>0 such that

|φ(y)φ(y)|CinfYjφ{d(y,y)+d(Fy,Fy)+γs(y,y)}for all y,yYjj1.𝜑𝑦𝜑superscript𝑦𝐶subscriptinfimumsubscript𝑌𝑗𝜑𝑑𝑦superscript𝑦𝑑𝐹𝑦𝐹superscript𝑦superscript𝛾𝑠𝑦superscript𝑦for all y,yYjj1.|\varphi(y)-\varphi(y^{\prime})|\leq C{\textstyle\inf_{Y_{j}}}\varphi\{d(y,y^{\prime})+d(Fy,Fy^{\prime})+\gamma^{s(y,y^{\prime})}\}\quad\text{for all $y,y^{\prime}\in Y_{j}$, $j\geq 1$.}
Proof.

Let y~=Y~Ws(y)~𝑦~𝑌superscript𝑊𝑠𝑦\tilde{y}={\widetilde{Y}}\cap W^{s}(y), y~=Y~Ws(y)superscript~𝑦~𝑌superscript𝑊𝑠superscript𝑦\tilde{y}^{\prime}={\widetilde{Y}}\cap W^{s}(y^{\prime}). Since φ~~𝜑\tilde{\varphi} is constant along stable leaves, it follows as in the proof of Proposition 6.1 that

|φ~(y)φ~(y)|=|φ~(y~)φ~(y~)|infYjφγs(y~,y~)=infYjφγs(y,y).~𝜑𝑦~𝜑superscript𝑦~𝜑~𝑦~𝜑superscript~𝑦much-less-thansubscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠~𝑦superscript~𝑦subscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠𝑦superscript𝑦|\tilde{\varphi}(y)-\tilde{\varphi}(y^{\prime})|=|\tilde{\varphi}(\tilde{y})-\tilde{\varphi}(\tilde{y}^{\prime})|\ll{\textstyle\inf_{Y_{j}}}\varphi\,\gamma^{s(\tilde{y},\tilde{y}^{\prime})}={\textstyle\inf_{Y_{j}}}\varphi\,\gamma^{s(y,y^{\prime})}.

Hence by (6.2) and (H)(b)

|φ(y)φ(y)|𝜑𝑦𝜑superscript𝑦\displaystyle|\varphi(y)-\varphi(y^{\prime})| |φ~(y)φ~(y)|+|χ(Fy)χ(Fy)|+|χ(y)χ(y)|absent~𝜑𝑦~𝜑superscript𝑦𝜒𝐹𝑦𝜒𝐹superscript𝑦𝜒𝑦𝜒superscript𝑦\displaystyle\leq|\tilde{\varphi}(y)-\tilde{\varphi}(y^{\prime})|+|\chi(Fy)-\chi(Fy^{\prime})|+|\chi(y)-\chi(y^{\prime})|
infYjφ{γs(y,y)+d(Fy,Fy)+γs(Fy,Fy)+d(y,y)}.much-less-thanabsentsubscriptinfimumsubscript𝑌𝑗𝜑superscript𝛾𝑠𝑦superscript𝑦𝑑𝐹𝑦𝐹superscript𝑦superscript𝛾𝑠𝐹𝑦𝐹superscript𝑦𝑑𝑦superscript𝑦\displaystyle\ll{\textstyle\inf_{Y_{j}}}\varphi\{\gamma^{s(y,y^{\prime})}+d(Fy,Fy^{\prime})+\gamma^{s(Fy,Fy^{\prime})}+d(y,y^{\prime})\}.

The result follows since γs(Fy,Fy)=γ1γs(y,y)superscript𝛾𝑠𝐹𝑦𝐹superscript𝑦superscript𝛾1superscript𝛾𝑠𝑦superscript𝑦\gamma^{s(Fy,Fy^{\prime})}=\gamma^{-1}\gamma^{s(y,y^{\prime})}. ∎

Next, we relate the two suspension flows Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} and F~t:Yφ~Yφ~:subscript~𝐹𝑡superscript𝑌~𝜑superscript𝑌~𝜑\widetilde{F}_{t}:Y^{\tilde{\varphi}}\to Y^{\tilde{\varphi}}. Note that (y,φ(y))𝑦𝜑𝑦(y,\varphi(y)) is identified with (Fy,0)𝐹𝑦0(Fy,0) in the first flow and (y,φ~(y))𝑦~𝜑𝑦(y,\tilde{\varphi}(y)) is identified with (Fy,0)𝐹𝑦0(Fy,0) in the second flow. Define

g+:YφYφ~,g+(y,u)=(y,u+χ(y)+|χ|),:subscript𝑔formulae-sequencesuperscript𝑌𝜑superscript𝑌~𝜑subscript𝑔𝑦𝑢𝑦𝑢𝜒𝑦subscript𝜒\displaystyle g_{+}:Y^{\varphi}\to Y^{\tilde{\varphi}},\qquad g_{+}(y,u)=(y,u+\chi(y)+|\chi|_{\infty}),
g:Yφ~Yφ,g(y,u)=(y,uχ(y)+|χ|),:subscript𝑔formulae-sequencesuperscript𝑌~𝜑superscript𝑌𝜑subscript𝑔𝑦𝑢𝑦𝑢𝜒𝑦subscript𝜒\displaystyle g_{-}:Y^{\tilde{\varphi}}\to Y^{\varphi},\qquad g_{-}(y,u)=(y,u-\chi(y)+|\chi|_{\infty}),

computed modulo identifications. Using (6.2) and the identifications on Yφ~superscript𝑌~𝜑Y^{\tilde{\varphi}},

g+(y,φ(y))=(y,φ(y)+χ(y)+|χ|)subscript𝑔𝑦𝜑𝑦𝑦𝜑𝑦𝜒𝑦subscript𝜒\displaystyle g_{+}(y,\varphi(y))=(y,\varphi(y)+\chi(y)+|\chi|_{\infty}) =(y,φ~(y)+χ(Fy)+|χ|)absent𝑦~𝜑𝑦𝜒𝐹𝑦subscript𝜒\displaystyle=(y,\tilde{\varphi}(y)+\chi(Fy)+|\chi|_{\infty})
(Fy,χ(Fy)+|χ|)=g+(Fy,0),similar-toabsent𝐹𝑦𝜒𝐹𝑦subscript𝜒subscript𝑔𝐹𝑦0\displaystyle\sim(Fy,\chi(Fy)+|\chi|_{\infty})=g_{+}(Fy,0),

so g+subscript𝑔g_{+} respects the identification on Yφsuperscript𝑌𝜑Y^{\varphi} and hence is well-defined. It follows easily that g+:YφYφ~:subscript𝑔superscript𝑌𝜑superscript𝑌~𝜑g_{+}:Y^{\varphi}\to Y^{\tilde{\varphi}} is a measure-preserving semiconjugacy between the two suspension flows. Similarly, gsubscript𝑔g_{-} is well-defined and gg+=F2|χ|:YφYφ:subscript𝑔subscript𝑔subscript𝐹2subscript𝜒superscript𝑌𝜑superscript𝑌𝜑g_{-}\circ g_{+}=F_{2|\chi|_{\infty}}:Y^{\varphi}\to Y^{\varphi}.

Given observables v,w:Yφ:𝑣𝑤superscript𝑌𝜑v,w:Y^{\varphi}\to{\mathbb{R}}, let v~=vg,w~=wg:Yφ~:formulae-sequence~𝑣𝑣subscript𝑔~𝑤𝑤subscript𝑔superscript𝑌~𝜑\tilde{v}=v\circ g_{-},\,\tilde{w}=w\circ g_{-}:Y^{\tilde{\varphi}}\to{\mathbb{R}}. When speaking of γ(Yφ~)subscript𝛾superscript𝑌~𝜑{\mathcal{H}}_{\gamma}(Y^{\tilde{\varphi}}) and so on, we use the metric d1(y,y)=d(y,y)ηsubscript𝑑1𝑦superscript𝑦𝑑superscript𝑦superscript𝑦𝜂d_{1}(y,y^{\prime})=d(y,y^{\prime})^{\eta} on Y𝑌Y instead of d𝑑d. Let γ1=γηsubscript𝛾1superscript𝛾𝜂\gamma_{1}=\gamma^{\eta}.

Let γ,η(Yφ)={v:Yφ:vγ,η<}superscriptsubscript𝛾𝜂superscript𝑌𝜑conditional-set𝑣:superscript𝑌𝜑subscriptsuperscriptnorm𝑣𝛾𝜂{\mathcal{H}}_{\gamma,\eta}^{*}(Y^{\varphi})=\{v:Y^{\varphi}\to{\mathbb{R}}:\|v\|^{*}_{\gamma,\eta}<\infty\} and γ,0,m(Yφ)={w:Yφ:wγ,0,m<}superscriptsubscript𝛾0𝑚superscript𝑌𝜑conditional-set𝑤:superscript𝑌𝜑subscriptsuperscriptnorm𝑤𝛾0𝑚{\mathcal{H}}_{\gamma,0,m}^{*}(Y^{\varphi})=\{w:Y^{\varphi}\to{\mathbb{R}}:\|w\|^{*}_{\gamma,0,m}<\infty\} where

vγ,η=vγ,η+vF2|χ|γ,η,wγ,0,m=wγ,0,m+wF2|χ|γ,0,m.formulae-sequencesubscriptsuperscriptnorm𝑣𝛾𝜂subscriptnorm𝑣𝛾𝜂subscriptnorm𝑣subscript𝐹2subscript𝜒𝛾𝜂subscriptsuperscriptnorm𝑤𝛾0𝑚subscriptnorm𝑤𝛾0𝑚subscriptnorm𝑤subscript𝐹2subscript𝜒𝛾0𝑚\|v\|^{*}_{\gamma,\eta}=\|v\|_{\gamma,\eta}+\|v\circ F_{2|\chi|_{\infty}}\|_{\gamma,\eta},\qquad\|w\|^{*}_{\gamma,0,m}=\|w\|_{\gamma,0,m}+\|w\circ F_{2|\chi|_{\infty}}\|_{\gamma,0,m}.
Lemma 6.3

Let vγ,η(Yφ)𝑣subscriptsuperscript𝛾𝜂superscript𝑌𝜑v\in{\mathcal{H}}^{*}_{\gamma,\eta}(Y^{\varphi}), wγ,0,m(Yφ)𝑤subscriptsuperscript𝛾0𝑚superscript𝑌𝜑w\in{\mathcal{H}}^{*}_{\gamma,0,m}(Y^{\varphi}), for some m1𝑚1m\geq 1. Then v~γ1,η(Yφ~)~𝑣subscriptsubscript𝛾1𝜂superscript𝑌~𝜑\tilde{v}\in{\mathcal{H}}_{\gamma_{1},\eta}(Y^{\tilde{\varphi}}), w~γ1,0,m(Yφ~)~𝑤subscriptsubscript𝛾10𝑚superscript𝑌~𝜑\tilde{w}\in{\mathcal{H}}_{\gamma_{1},0,m}(Y^{\tilde{\varphi}}), and v~γ1,η4C3vγ,ηsubscriptnorm~𝑣subscript𝛾1𝜂4subscript𝐶3subscriptsuperscriptnorm𝑣𝛾𝜂\|\tilde{v}\|_{\gamma_{1},\eta}\leq 4C_{3}\|v\|^{*}_{\gamma,\eta}, w~γ1,0,m2C3wγ,0,msubscriptnorm~𝑤subscript𝛾10𝑚2subscript𝐶3subscriptsuperscriptnorm𝑤𝛾0𝑚\|\tilde{w}\|_{\gamma_{1},0,m}\leq 2C_{3}\|w\|^{*}_{\gamma,0,m}.

Proof.

We have v~(y,u)=v(y,uχ(y)+|χ|)~𝑣𝑦𝑢𝑣𝑦𝑢𝜒𝑦subscript𝜒\tilde{v}(y,u)=v(y,u-\chi(y)+|\chi|_{\infty}). It is immediate that |v~||v|subscript~𝑣subscript𝑣|\tilde{v}|_{\infty}\leq|v|_{\infty}.

Now let (y,u),(y,u)Yφ~𝑦𝑢superscript𝑦𝑢superscript𝑌~𝜑(y,u),\,(y^{\prime},u)\in Y^{\tilde{\varphi}}. Suppose without loss that χ(y)χ(y)𝜒𝑦𝜒superscript𝑦\chi(y)\geq\chi(y^{\prime}). First, we consider the case uχ(y)+|χ|φ(y)𝑢𝜒𝑦subscript𝜒𝜑𝑦u-\chi(y)+|\chi|_{\infty}\leq\varphi(y), uχ(y)+|χ|φ(y)𝑢𝜒superscript𝑦subscript𝜒𝜑superscript𝑦u-\chi(y^{\prime})+|\chi|_{\infty}\leq\varphi(y^{\prime}). By (H)(b) and the definition of vγ,ηsubscriptnorm𝑣𝛾𝜂\|v\|_{\gamma,\eta},

|v~(y,u)v~(y,u)|~𝑣𝑦𝑢~𝑣superscript𝑦𝑢\displaystyle|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)| |v(y,uχ(y)+|χ|)v(y,uχ(y)+|χ|)|absent𝑣𝑦𝑢𝜒𝑦subscript𝜒𝑣superscript𝑦𝑢𝜒𝑦subscript𝜒\displaystyle\leq\big{|}v(y,u-\chi(y)+|\chi|_{\infty})-v(y^{\prime},u-\chi(y)+|\chi|_{\infty})\big{|}
+|v(y,uχ(y)+|χ|)v(y,uχ(y)+|χ|)|𝑣superscript𝑦𝑢𝜒𝑦subscript𝜒𝑣superscript𝑦𝑢𝜒superscript𝑦subscript𝜒\displaystyle\qquad+\big{|}v(y^{\prime},u-\chi(y)+|\chi|_{\infty})-v(y^{\prime},u-\chi(y^{\prime})+|\chi|_{\infty})\big{|}
|v|γφ(y)(d(y,y)+γs(y,y))+|v|,η|χ(y)χ(y)|ηabsentsubscript𝑣𝛾𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦subscript𝑣𝜂superscript𝜒𝑦𝜒superscript𝑦𝜂\displaystyle\leq|v|_{\gamma}\varphi(y)(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})})+|v|_{\infty,\eta}|\chi(y)-\chi(y^{\prime})|^{\eta}
2|v|γφ~(y)(d(y,y)+γs(y,y))+|v|,ηC3(d(y,y)+γs(y,y))ηabsent2subscript𝑣𝛾~𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦subscript𝑣𝜂subscript𝐶3superscript𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝜂\displaystyle\leq 2|v|_{\gamma}\tilde{\varphi}(y)(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})})+|v|_{\infty,\eta}C_{3}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})})^{\eta}
2C3vγ,ηφ~(y)(d1(y,y)+γ1s(y,y)).absent2subscript𝐶3subscriptnorm𝑣𝛾𝜂~𝜑𝑦subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦\displaystyle\leq 2C_{3}\|v\|_{\gamma,\eta}\tilde{\varphi}(y)(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}).

Second, we consider the case uχ(y)+|χ|χ(y)+|χ|𝑢𝜒𝑦subscript𝜒𝜒superscript𝑦subscript𝜒u\geq\chi(y)+|\chi|_{\infty}\geq\chi(y^{\prime})+|\chi|_{\infty}. Then we can write g(y,u)=Fσ(y,uχ(y)|χ|)subscript𝑔𝑦𝑢subscript𝐹𝜎𝑦𝑢𝜒𝑦subscript𝜒g_{-}(y,u)=F_{\sigma}(y,u-\chi(y)-|\chi|_{\infty}), g(y,u)=Fσ(y,uχ(y)|χ|)subscript𝑔superscript𝑦𝑢subscript𝐹𝜎superscript𝑦𝑢𝜒superscript𝑦subscript𝜒g_{-}(y^{\prime},u)=F_{\sigma}(y^{\prime},u-\chi(y^{\prime})-|\chi|_{\infty}) where σ=2|χ|𝜎2subscript𝜒\sigma=2|\chi|_{\infty}, so

|v~(y,u)v~(y,u)|=|vFσ(y,uχ(y)|χ|)vFσ(y,uχ(y)|χ|)|.~𝑣𝑦𝑢~𝑣superscript𝑦𝑢𝑣subscript𝐹𝜎𝑦𝑢𝜒𝑦subscript𝜒𝑣subscript𝐹𝜎superscript𝑦𝑢𝜒superscript𝑦subscript𝜒|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|=\big{|}v\circ F_{\sigma}(y,u-\chi(y)-|\chi|_{\infty})-v\circ F_{\sigma}(y^{\prime},u-\chi(y^{\prime})-|\chi|_{\infty})\big{|}.

Proceeding as in the first case,

|v~(y,u)v~(y,u)|2C3vFσγ,ηφ~(y)(d1(y,y)+γ1s(y,y)).~𝑣𝑦𝑢~𝑣superscript𝑦𝑢2subscript𝐶3subscriptnorm𝑣subscript𝐹𝜎𝛾𝜂~𝜑𝑦subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|\leq 2C_{3}\|v\circ F_{\sigma}\|_{\gamma,\eta}\tilde{\varphi}(y)(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}).

This leaves the case u<χ(y)+|χ|2|χ|evaluated-at𝑢bralimit-from𝜒𝑦𝜒2subscript𝜒u<\chi(y)+|\chi|_{\infty}\leq 2|\chi|_{\infty} and umin{φ(y)+χ(y)|χ|,φ(y)+χ(y)|χ|}infφ2|χ|𝑢𝜑𝑦𝜒𝑦subscript𝜒𝜑superscript𝑦𝜒superscript𝑦subscript𝜒infimum𝜑2subscript𝜒u\geq\min\{\varphi(y)+\chi(y)-|\chi|_{\infty},\varphi(y^{\prime})+\chi(y^{\prime})-|\chi|_{\infty}\}\geq\inf\varphi-2|\chi|_{\infty}. This is impossible since infφ>4|χ|infimum𝜑4subscript𝜒\inf\varphi>4|\chi|_{\infty}. Hence

|v~(y,u)v~(y,u)|2C3vγ,ηφ~(y)(d1(y,y)+γ1s(y,y))for all (y,u),(y,u)Yφ~,~𝑣𝑦𝑢~𝑣superscript𝑦𝑢2subscript𝐶3subscriptsuperscriptnorm𝑣𝛾𝜂~𝜑𝑦subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦for all (y,u),(y,u)Yφ~|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|\leq 2C_{3}\|v\|^{*}_{\gamma,\eta}\tilde{\varphi}(y)(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})})\quad\text{for all $(y,u),(y^{\prime},u)\in Y^{\tilde{\varphi}}$},

so v~γ12C3vγ,ηsubscriptnorm~𝑣subscript𝛾12subscript𝐶3subscriptsuperscriptnorm𝑣𝛾𝜂\|\tilde{v}\|_{\gamma_{1}}\leq 2C_{3}\|v\|^{*}_{\gamma,\eta}.

The estimate for |v~|,ηsubscript~𝑣𝜂|\tilde{v}|_{\infty,\eta} splits into cases similarly. Let 0u<uφ~(y)0𝑢superscript𝑢~𝜑𝑦0\leq u<u^{\prime}\leq\tilde{\varphi}(y). Then

|v~(y,u)v~(y,u)|{|v|,η|uu|ηuχ(y)+|χ|φ(y)|vFσ|,η|uu|ηuχ(y)+|χ|.~𝑣𝑦𝑢~𝑣𝑦superscript𝑢casessubscript𝑣𝜂superscript𝑢superscript𝑢𝜂superscript𝑢𝜒𝑦subscript𝜒𝜑𝑦subscript𝑣subscript𝐹𝜎𝜂superscript𝑢superscript𝑢𝜂𝑢𝜒𝑦subscript𝜒|\tilde{v}(y,u)-\tilde{v}(y,u^{\prime})|\leq\begin{cases}|v|_{\infty,\eta}|u-u^{\prime}|^{\eta}&u^{\prime}-\chi(y)+|\chi|_{\infty}\leq\varphi(y)\\ |v\circ F_{\sigma}|_{\infty,\eta}|u-u^{\prime}|^{\eta}&u\geq\chi(y)+|\chi|_{\infty}\end{cases}.

This leaves the case uχ(y)+|χ|>φ(y)superscript𝑢𝜒𝑦subscript𝜒𝜑𝑦u^{\prime}-\chi(y)+|\chi|_{\infty}>\varphi(y) and u<χ(y)+|χ|evaluated-at𝑢bralimit-from𝜒𝑦𝜒u<\chi(y)+|\chi|_{\infty}. But then uu>φ(y)+2χ(y)>φ(y)2|χ|>12φ(y)12superscript𝑢𝑢𝜑𝑦2𝜒𝑦𝜑𝑦2subscript𝜒12𝜑𝑦12u^{\prime}-u>\varphi(y)+2\chi(y)>\varphi(y)-2|\chi|_{\infty}>\frac{1}{2}\varphi(y)\geq\frac{1}{2}, so we obtain |v~(y,u)v~(y,u)|2|v|4|v||uu|η~𝑣𝑦𝑢~𝑣𝑦superscript𝑢2subscript𝑣4subscript𝑣superscript𝑢superscript𝑢𝜂|\tilde{v}(y,u)-\tilde{v}(y,u^{\prime})|\leq 2|v|_{\infty}\leq 4|v|_{\infty}|u-u^{\prime}|^{\eta}. Hence |v~|,η4vγ,ηsubscript~𝑣𝜂4subscriptsuperscriptnorm𝑣𝛾𝜂|\tilde{v}|_{\infty,\eta}\leq 4\|v\|^{*}_{\gamma,\eta} completing the estimate for v~γ1,ηsubscriptnorm~𝑣subscript𝛾1𝜂\|\tilde{v}\|_{\gamma_{1},\eta}. The calculation for w~~𝑤\tilde{w} is similar.  ∎

We say that a Gibbs-Markov flow has approximate eigenfunctions if this is the case for F~tsubscript~𝐹𝑡\widetilde{F}_{t} (equivalently F¯tsubscript¯𝐹𝑡\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu_{t}).

Theorem 6.4

Suppose that Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow such that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) for some β>1𝛽1\beta>1. Assume absence of approximate eigenfunctions. Then there exists m1𝑚1m\geq 1 and C>0𝐶0C>0 such that

|ρv,w(t)|Cvγ,ηwγ,0,mt(β1)for all vγ,η(Yφ)wγ,0,m(Yφ)t>1.subscript𝜌𝑣𝑤𝑡𝐶subscriptsuperscriptnorm𝑣𝛾𝜂subscriptsuperscriptnorm𝑤𝛾0𝑚superscript𝑡𝛽1for all vγ,η(Yφ)wγ,0,m(Yφ)t>1|\rho_{v,w}(t)|\leq C\|v\|^{*}_{\gamma,\eta}\|w\|^{*}_{\gamma,0,m}\,t^{-(\beta-1)}\quad\text{for all $v\in{\mathcal{H}}^{*}_{\gamma,\eta}(Y^{\varphi})$, $w\in{\mathcal{H}}^{*}_{\gamma,0,m}(Y^{\varphi})$, $t>1$}.
Proof.

Since g+subscript𝑔g_{+} is a measure-preserving semiconjugacy and gg+=F2|χ|subscript𝑔subscript𝑔subscript𝐹2subscript𝜒g_{-}\circ g_{+}=F_{2|\chi|_{\infty}},

YφvwFt𝑑μφsubscriptsuperscript𝑌𝜑𝑣𝑤subscript𝐹𝑡differential-dsuperscript𝜇𝜑\displaystyle\int_{Y^{\varphi}}v\,w\circ F_{t}\,d\mu^{\varphi} =Yφvgg+wgg+Ft𝑑μφabsentsubscriptsuperscript𝑌𝜑𝑣subscript𝑔subscript𝑔𝑤subscript𝑔subscript𝑔subscript𝐹𝑡differential-dsuperscript𝜇𝜑\displaystyle=\int_{Y^{\varphi}}v\circ g_{-}\circ g_{+}\,w\circ g_{-}\circ g_{+}\circ F_{t}\,d\mu^{\varphi}
=Yφv~g+w~F~tg+𝑑μφ=Yφ~v~w~F~t𝑑μφ~absentsubscriptsuperscript𝑌𝜑~𝑣subscript𝑔~𝑤subscript~𝐹𝑡subscript𝑔differential-dsuperscript𝜇𝜑subscriptsuperscript𝑌~𝜑~𝑣~𝑤subscript~𝐹𝑡differential-dsuperscript𝜇~𝜑\displaystyle=\int_{Y^{\varphi}}\tilde{v}\circ g_{+}\,\tilde{w}\circ\widetilde{F}_{t}\circ g_{+}\,d\mu^{\varphi}=\int_{Y^{\tilde{\varphi}}}\tilde{v}\,\tilde{w}\circ\widetilde{F}_{t}\,d\mu^{\tilde{\varphi}}

where F~tsubscript~𝐹𝑡\widetilde{F}_{t} does not possess approximate eigenfunctions. Note also that μ(φ~>t)=O(tβ)𝜇~𝜑𝑡𝑂superscript𝑡𝛽\mu(\tilde{\varphi}>t)=O(t^{-\beta}). By Lemma 6.3, v~γ1,η(Yφ~)~𝑣subscriptsubscript𝛾1𝜂superscript𝑌~𝜑\tilde{v}\in{\mathcal{H}}_{\gamma_{1},\eta}(Y^{\tilde{\varphi}}), w~γ1,0,m(Yφ~)~𝑤subscriptsubscript𝛾10𝑚superscript𝑌~𝜑\tilde{w}\in{\mathcal{H}}_{\gamma_{1},0,m}(Y^{\tilde{\varphi}}).

By Theorem 3.2, we can choose m1𝑚1m\geq 1 such that |ρv,w(t)|=|Yφ~v~w~F~t𝑑μφ~Yφ~v~𝑑μφ~Yφ~w~𝑑μφ~|v~γ1,ηw~γ1,0,mt(β1)8C32vγ,ηwγ,0,mt(β1)subscript𝜌𝑣𝑤𝑡subscriptsuperscript𝑌~𝜑~𝑣~𝑤subscript~𝐹𝑡differential-dsuperscript𝜇~𝜑subscriptsuperscript𝑌~𝜑~𝑣differential-dsuperscript𝜇~𝜑subscriptsuperscript𝑌~𝜑~𝑤differential-dsuperscript𝜇~𝜑much-less-thansubscriptnorm~𝑣subscript𝛾1𝜂subscriptnorm~𝑤subscript𝛾10𝑚superscript𝑡𝛽18superscriptsubscript𝐶32subscriptsuperscriptnorm𝑣𝛾𝜂subscriptsuperscriptnorm𝑤𝛾0𝑚superscript𝑡𝛽1|\rho_{v,w}(t)|=|\int_{Y^{\tilde{\varphi}}}\tilde{v}\,\tilde{w}\circ\widetilde{F}_{t}\,d\mu^{\tilde{\varphi}}-\int_{Y^{\tilde{\varphi}}}\tilde{v}\,d\mu^{\tilde{\varphi}}\int_{Y^{\tilde{\varphi}}}\tilde{w}\,d\mu^{\tilde{\varphi}}|\ll\|\tilde{v}\|_{\gamma_{1},\eta}\|\tilde{w}\|_{\gamma_{1},0,m}\,t^{-(\beta-1)}\leq 8C_{3}^{2}\|v\|^{*}_{\gamma,\eta}\|w\|^{*}_{\gamma,0,m}\,t^{-(\beta-1)}. ∎

6.2 Periodic data and absence of approximate eigenfunctions

In this subsection, we recall the relationship between periodic data and approximate eigenfunctions and review two sufficient conditions to rule out the existence of approximate eigenfunctions. We continue to assume that Ftsubscript𝐹𝑡F_{t} is a Gibbs-Markov flow as in Subsection 6.1.

Define φn=j=0n1φFjsubscript𝜑𝑛superscriptsubscript𝑗0𝑛1𝜑superscript𝐹𝑗\varphi_{n}=\sum_{j=0}^{n-1}\varphi\circ F^{j}. Similarly, define φ~nsubscript~𝜑𝑛\tilde{\varphi}_{n} and φ¯nsubscript¯𝜑𝑛{\bar{\varphi}}_{n}. If y𝑦y is a periodic point of period p𝑝p for F𝐹F (that is, Fpy=ysuperscript𝐹𝑝𝑦𝑦F^{p}y=y), then y𝑦y is periodic of period =φp(y)subscript𝜑𝑝𝑦{\mathcal{L}}=\varphi_{p}(y) for Ftsubscript𝐹𝑡F_{t} (that is, Fy=ysubscript𝐹𝑦𝑦F_{\mathcal{L}}y=y). Recall that π¯:YY¯:¯𝜋𝑌¯𝑌\bar{\pi}:Y\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is the quotient projection.

Proposition 6.5

Suppose that there exist approximate eigenfunctions on Z0Y¯subscript𝑍0¯𝑌Z_{0}\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. Let α,C,bk,nk𝛼𝐶subscript𝑏𝑘subscript𝑛𝑘\alpha,C,b_{k},n_{k} be as in Definition 2.3. If yπ¯1Z0𝑦superscript¯𝜋1subscript𝑍0y\in\bar{\pi}^{-1}Z_{0} is a periodic point with Fpy=ysuperscript𝐹𝑝𝑦𝑦F^{p}y=y and Fy=ysubscript𝐹𝑦𝑦F_{\mathcal{L}}y=y where =φp(y)subscript𝜑𝑝𝑦{\mathcal{L}}=\varphi_{p}(y), then

dist(bknkpψk,2π)C(infφ)1|bk|αfor all k1.distsubscript𝑏𝑘subscript𝑛𝑘𝑝subscript𝜓𝑘2𝜋𝐶superscriptinfimum𝜑1superscriptsubscript𝑏𝑘𝛼for all k1\operatorname{dist}(b_{k}n_{k}{\mathcal{L}}-p\psi_{k},2\pi{\mathbb{Z}})\leq C(\inf\varphi)^{-1}{\mathcal{L}}|b_{k}|^{-\alpha}\quad\text{for all $k\geq 1$}. (6.3)
Proof.

Define y¯=π¯yZ0¯𝑦¯𝜋𝑦subscript𝑍0\bar{y}=\bar{\pi}y\in Z_{0} and note that F¯py¯=F¯pπ¯y=π¯Fpy=y¯superscript¯𝐹𝑝¯𝑦superscript¯𝐹𝑝¯𝜋𝑦¯𝜋superscript𝐹𝑝𝑦¯𝑦\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{p}\bar{y}=\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{p}\bar{\pi}y=\bar{\pi}F^{p}y=\bar{y}. By (6.2),

φ¯p(y¯)=φ~p(y)=φp(y)+χ(y)χ(Fpy)=.subscript¯𝜑𝑝¯𝑦subscript~𝜑𝑝𝑦subscript𝜑𝑝𝑦𝜒𝑦𝜒superscript𝐹𝑝𝑦\displaystyle{\bar{\varphi}}_{p}(\bar{y})=\tilde{\varphi}_{p}(y)=\varphi_{p}(y)+\chi(y)-\chi(F^{p}y)={\mathcal{L}}.

Now (Mbpv)(y¯)=eibφ¯p(y¯)v(F¯py¯)=eibv(y¯)superscriptsubscript𝑀𝑏𝑝𝑣¯𝑦superscript𝑒𝑖𝑏subscript¯𝜑𝑝¯𝑦𝑣superscript¯𝐹𝑝¯𝑦superscript𝑒𝑖𝑏𝑣¯𝑦(M_{b}^{p}v)(\bar{y})=e^{ib{\bar{\varphi}}_{p}(\bar{y})}v(\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu^{p}\bar{y})=e^{ib{\mathcal{L}}}v(\bar{y}). Hence substituting y¯¯𝑦\bar{y} into (2.2), we obtain |eibknkeipψk|Cp|bk|αsuperscript𝑒𝑖subscript𝑏𝑘subscript𝑛𝑘superscript𝑒𝑖𝑝subscript𝜓𝑘𝐶𝑝superscriptsubscript𝑏𝑘𝛼|e^{ib_{k}n_{k}{\mathcal{L}}}-e^{ip\psi_{k}}|\leq Cp|b_{k}|^{-\alpha}. Also =φp(y)pinfφsubscript𝜑𝑝𝑦𝑝infimum𝜑{\mathcal{L}}=\varphi_{p}(y)\geq p\inf\varphi. ∎

The following Diophantine condition is based on [18, Section 13]. (Unlike in [18], we have to consider periods corresponding to three periodic points instead of two.)

Proposition 6.6

Let y1,y2,y3Yjsubscript𝑦1subscript𝑦2subscript𝑦3subscript𝑌𝑗y_{1},y_{2},y_{3}\in\bigcup Y_{j} be fixed points for F𝐹F, and let i=φ(yi)subscript𝑖𝜑subscript𝑦𝑖{\mathcal{L}}_{i}=\varphi(y_{i}), i=1,2,3𝑖123i=1,2,3, be the corresponding periods for Ftsubscript𝐹𝑡F_{t}. Let Z0Y¯subscript𝑍0¯𝑌Z_{0}\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} be the finite subsystem corresponding to the three partition elements containing π¯y1,π¯y2,π¯y3¯𝜋subscript𝑦1¯𝜋subscript𝑦2¯𝜋subscript𝑦3\bar{\pi}y_{1},\bar{\pi}y_{2},\bar{\pi}y_{3}.

If (13)/(23)subscript1subscript3subscript2subscript3({\mathcal{L}}_{1}-{\mathcal{L}}_{3})/({\mathcal{L}}_{2}-{\mathcal{L}}_{3}) is Diophantine, then there do not exist approximate eigenfunctions on Z0subscript𝑍0Z_{0}.

Proof.

Using Proposition 6.5, the proof is identical to that of [26, Proposition 5.3]. ∎

The condition in Proposition 6.6 is satisfied with probability one but is not robust. Using the notion of good asymptotics [19], we obtain an open and dense condition.

Proposition 6.7

Let Z0Y¯subscript𝑍0¯𝑌Z_{0}\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} be a finite subsystem. Let y0π¯1Z0subscript𝑦0superscript¯𝜋1subscript𝑍0y_{0}\in\bar{\pi}^{-1}Z_{0} be a fixed point for F𝐹F with period 0=φ(y0)subscript0𝜑subscript𝑦0{\mathcal{L}}_{0}=\varphi(y_{0}) for the flow. Let yNπ¯1Z0subscript𝑦𝑁superscript¯𝜋1subscript𝑍0y_{N}\in\bar{\pi}^{-1}Z_{0}, N1𝑁1N\geq 1, be a sequence of periodic points with FNyN=yNsuperscript𝐹𝑁subscript𝑦𝑁subscript𝑦𝑁F^{N}y_{N}=y_{N} such that their periods N=φN(yN)subscript𝑁subscript𝜑𝑁subscript𝑦𝑁{\mathcal{L}}_{N}=\varphi_{N}(y_{N}) for the flow Ftsubscript𝐹𝑡F_{t} satisfy

N=N0+κ+ENγNcos(Nω+ωN)+o(γN),subscript𝑁𝑁subscript0𝜅subscript𝐸𝑁superscript𝛾𝑁𝑁𝜔subscript𝜔𝑁𝑜superscript𝛾𝑁{\mathcal{L}}_{N}=N{\mathcal{L}}_{0}+\kappa+E_{N}\gamma^{N}\cos(N\omega+\omega_{N})+o(\gamma^{N}),

where κ𝜅\kappa\in{\mathbb{R}}, γ(0,1)𝛾01\gamma\in(0,1) are constants, ENsubscript𝐸𝑁E_{N}\in{\mathbb{R}} is a bounded sequence with lim infN|EN|>0subscriptlimit-infimum𝑁subscript𝐸𝑁0\liminf_{N\to\infty}|E_{N}|>0, and either (i) ω=0𝜔0\omega=0 and ωN0subscript𝜔𝑁0\omega_{N}\equiv 0, or (ii) ω(0,π)𝜔0𝜋\omega\in(0,\pi) and ωN(ω0π/12,ω0+π/12)subscript𝜔𝑁subscript𝜔0𝜋12subscript𝜔0𝜋12\omega_{N}\in(\omega_{0}-\pi/12,\omega_{0}+\pi/12) for some ω0subscript𝜔0\omega_{0}. (Such a sequence of periodic points is said to have good asymptotics.)

Then there do not exist approximate eigenfunctions on Z0subscript𝑍0Z_{0}.

Proof.

Using Proposition 6.5, the proof is identical to that of [26, Proposition 5.5]. ∎

By [19], for any finite subsystem Z0subscript𝑍0Z_{0}, the existence of periodic points with good asymptotics in π¯1Z0superscript¯𝜋1subscript𝑍0\bar{\pi}^{-1}Z_{0} is a C2superscript𝐶2C^{2}-open and Csuperscript𝐶C^{\infty}-dense condition. Although [19] is set in the uniformly hyperbolic setting, the construction applies directly to the current set up as we now explain. Assume that (Y,d)𝑌𝑑(Y,d) is a Riemannian manifold. Let Z¯1subscript¯𝑍1\bar{Z}_{1} and Z¯2subscript¯𝑍2\bar{Z}_{2} be two of the partition elements in Z𝑍Z and set Zj=Intπ¯1Z¯jsubscript𝑍𝑗Intsuperscript¯𝜋1subscript¯𝑍𝑗Z_{j}=\operatorname{Int}\bar{\pi}^{-1}\bar{Z}_{j} for j=1,2𝑗12j=1,2. Assume that Z1subscript𝑍1Z_{1}, Z2subscript𝑍2Z_{2} are submanifolds of Y𝑌Y and that F𝐹F and φ𝜑\varphi are Crsuperscript𝐶𝑟C^{r} when restricted to Z1Z2subscript𝑍1subscript𝑍2Z_{1}\cup Z_{2} for some r2𝑟2r\geq 2.

Let y0Z1subscript𝑦0subscript𝑍1y_{0}\in Z_{1} be a fixed point for F𝐹F and choose a transverse homoclinic point in Z2subscript𝑍2Z_{2}. Following [19], we construct a sequence of N𝑁N-periodic points yNsubscript𝑦𝑁y_{N}, N1𝑁1N\geq 1, for F𝐹F with orbits lying in Z1Z2subscript𝑍1subscript𝑍2Z_{1}\cup Z_{2}. The sequence automatically has good asymptotics except that in exceptional cases it may be that lim infN|EN|=0subscriptlimit-infimum𝑁subscript𝐸𝑁0\liminf_{N\to\infty}|E_{N}|=0. By [19], the liminf is positive for a C2superscript𝐶2C^{2} open and Crsuperscript𝐶𝑟C^{r} dense set of roof functions φ𝜑\varphi.

Combining this construction with Proposition 6.7, it follows that nonexistence of approximate eigenfunctions holds for an open and dense set of smooth Gibbs-Markov flows.

Part II Mixing rates for nonuniformly hyperbolic flows

In this part of the paper, we show how the results for suspension flows in Part I can be translated into results for nonuniformly hyperbolic flows defined on an ambient manifold. In Section 7, we show how this is done under the assumption that condition (H) from Section 6 is valid. In Section 8, we describe a number of situations where condition (H) is satisfied. This includes all the examples considered here and in [26]. In Section 9, we consider in detail the planar infinite horizon Lorentz gas.

7 Nonuniformly hyperbolic flows and suspension flows

In this section, we describe a class of nonuniformly hyperbolic flows Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M that have most of the properties required for Ttsubscript𝑇𝑡T_{t} to be modelled by a Gibbs-Markov flow. (The remaining property, condition (H) from Section 6, is considered in Section 8.)

In Subsection 7.1, we consider a class of nonuniformly hyperbolic transformations f:XX:𝑓𝑋𝑋f:X\to X modelled by a Young tower [30, 31], making explicit the conditions from [30] that are needed for this paper. In Subsection 7.2, we consider flows that are Hölder suspensions over such a map f𝑓f and show how to model them, subject to condition (H), by a Gibbs-Markov flow. In Subsection 7.3, we generalise the Hölder structures in Subsection 7.2 to ones that are dynamically Hölder.

In applications, f𝑓f is typically a first-hit Poincaré map for the flow Ttsubscript𝑇𝑡T_{t} and hence is invertible. Invertibility is used in Proposition A.1 but not elsewhere, so many of our results do not rely on injectivity of f𝑓f.

7.1 Nonuniformly hyperbolic transformations f:XX:𝑓𝑋𝑋f:X\to X

Let f:XX:𝑓𝑋𝑋f:X\to X be a measurable transformation defined on a metric space (X,d)𝑋𝑑(X,d) with diamX1diam𝑋1\operatorname{diam}X\leq 1. We suppose that f𝑓f is nonuniformly hyperbolic in the sense that it is modelled by a Young tower [30, 31]. We recall the metric parts of the theory; the differential geometry part leading to an SRB or physical measure does not play an important role here.

Product structure

Let Y𝑌Y be a measurable subset of X𝑋X. Let 𝒲ssuperscript𝒲𝑠{\mathcal{W}}^{s} be a collection of disjoint measurable subsets of X𝑋X (called “stable leaves”) and let 𝒲usuperscript𝒲𝑢{\mathcal{W}}^{u} be a collection of disjoint measurable subsets of X𝑋X (called “unstable leaves”) such that each collection covers Y𝑌Y. Given yY𝑦𝑌y\in Y, let Ws(y)superscript𝑊𝑠𝑦W^{s}(y) and Wu(y)superscript𝑊𝑢𝑦W^{u}(y) denote the stable and unstable leaves containing y𝑦y.

We assume that for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, the intersection Ws(y)Wu(y)superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦W^{s}(y)\cap W^{u}(y^{\prime}) consists of precisely one point, denoted z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}), and that zY𝑧𝑌z\in Y. Also we suppose there is a constant C41subscript𝐶41C_{4}\geq 1 such that

d(y,z)C4d(y,y)for all y,yYz=Ws(y)Wu(y).𝑑𝑦𝑧subscript𝐶4𝑑𝑦superscript𝑦for all y,yYz=Ws(y)Wu(y).d(y,z)\leq C_{4}d(y,y^{\prime})\quad\text{for all $y,y^{\prime}\in Y$, $\,z=W^{s}(y)\cap W^{u}(y^{\prime})$.} (7.1)
Induced map

Next, let {Yj}subscript𝑌𝑗\{Y_{j}\} be an at most countable measurable partition of Y𝑌Y such that Yj=yYjWs(y)Ysubscript𝑌𝑗subscript𝑦subscript𝑌𝑗superscript𝑊𝑠𝑦𝑌Y_{j}=\bigcup_{y\in Y_{j}}W^{s}(y)\cap Y for all j1𝑗1j\geq 1. Also, fix τ:Y+:𝜏𝑌superscript\tau:Y\to{\mathbb{Z}}^{+} constant on partition elements such that fτ(y)yYsuperscript𝑓𝜏𝑦𝑦𝑌f^{\tau(y)}y\in Y for all yY𝑦𝑌y\in Y. Define F:YY:𝐹𝑌𝑌F:Y\to Y by Fy=fτ(y)y𝐹𝑦superscript𝑓𝜏𝑦𝑦Fy=f^{\tau(y)}y. Let μ𝜇\mu be an ergodic F𝐹F-invariant probability measure on Y𝑌Y and suppose that τ𝜏\tau is integrable. (It is not assumed that τ𝜏\tau is the first return time to Y𝑌Y.)

As in Section 3, we suppose that F(Ws(y))Ws(Fy)𝐹superscript𝑊𝑠𝑦superscript𝑊𝑠𝐹𝑦F(W^{s}(y))\subset W^{s}(Fy) for all yY𝑦𝑌y\in Y. Let Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} denote the space obtained from Y𝑌Y after quotienting by 𝒲ssuperscript𝒲𝑠{\mathcal{W}}^{s}, with natural projection π¯:YY¯:¯𝜋𝑌¯𝑌\bar{\pi}:Y\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. We assume that the quotient map F¯:Y¯Y¯:¯𝐹¯𝑌¯𝑌\mkern 3.5mu\overline{\mkern-3.5muF\mkern-0.5mu}\mkern 0.5mu:{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\to{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} is a Gibbs-Markov map as in Definition 2.1, with partition {Y¯j}subscript¯𝑌𝑗\{{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}\} and ergodic invariant probability measure μ¯=π¯μ¯𝜇subscript¯𝜋𝜇\bar{\mu}=\bar{\pi}_{*}\mu. Let s(y,y)𝑠𝑦superscript𝑦s(y,y^{\prime}) denote the separation time on Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}.

Contraction/expansion

Let Yj=π¯1Y¯jsubscript𝑌𝑗superscript¯𝜋1subscript¯𝑌𝑗Y_{j}=\bar{\pi}^{-1}{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}\!_{j}; these form a partition of Y𝑌Y and each Yjsubscript𝑌𝑗Y_{j} is a union of stable leaves. The separation time extends to Y𝑌Y, setting s(y,y)=s(π¯y,π¯y)𝑠𝑦superscript𝑦𝑠¯𝜋𝑦¯𝜋superscript𝑦s(y,y^{\prime})=s(\bar{\pi}y,\bar{\pi}y^{\prime}) for y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y.

We assume that there are constants C21subscript𝐶21C_{2}\geq 1, γ(0,1)𝛾01\gamma\in(0,1) such that for all n0𝑛0n\geq 0, y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y,

d(fny,fny)𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦\displaystyle d(f^{n}y,f^{n}y^{\prime}) C2γψn(y)d(y,y)absentsubscript𝐶2superscript𝛾subscript𝜓𝑛𝑦𝑑𝑦superscript𝑦\displaystyle\leq C_{2}\gamma^{\psi_{n}(y)}d(y,y^{\prime}) for all yWs(y),for all yWs(y)\displaystyle\quad\text{for all $y^{\prime}\in W^{s}(y)$}, (7.2)
d(fny,fny)𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦\displaystyle d(f^{n}y,f^{n}y^{\prime}) C2γs(y,y)ψn(y)absentsubscript𝐶2superscript𝛾𝑠𝑦superscript𝑦subscript𝜓𝑛𝑦\displaystyle\leq C_{2}\gamma^{s(y,y^{\prime})-\psi_{n}(y)} for all yWu(y),for all yWu(y)\displaystyle\quad\text{for all $y^{\prime}\in W^{u}(y)$}, (7.3)

where ψn(y)=#{j=1,,n:fjyY}subscript𝜓𝑛𝑦#conditional-set𝑗1𝑛superscript𝑓𝑗𝑦𝑌\psi_{n}(y)=\#\{j=1,\dots,n:f^{j}y\in Y\} is the number of returns of y𝑦y to Y𝑌Y by time n𝑛n. Note that conditions (3.1) and (3.2) are special cases of (7.2) and (7.3) where Y~~𝑌{\widetilde{Y}} can be chosen to be any fixed unstable leaf. In particular, all the conditions on F𝐹F in Sections 3 and 6 are satisfied.

In Sections 7.38.4 and 9, we make use of the condition

F(Wu(y)Yj)=Wu(Fy)Yfor all yYjj1.𝐹superscript𝑊𝑢𝑦subscript𝑌𝑗superscript𝑊𝑢𝐹𝑦𝑌for all yYjj1.F(W^{u}(y)\cap Y_{j})=W^{u}(Fy)\cap Y\quad\text{for all $y\in Y_{j}$, $j\geq 1$.} (7.4)
Remark 7.1

Further hypotheses in [30] ensure the existence of SRB measures on Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}, Y𝑌Y and X𝑋X. These assumptions are not required here and no special properties of μ𝜇\mu and μ¯¯𝜇\bar{\mu} (other than the properties mentioned above) are used.

Remark 7.2

The abstract setup in [30] essentially satisfies all of the assumptions above. However condition (7.2) is stated in the slightly weaker form d(fny,fny)C2γψn(y)𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦subscript𝐶2superscript𝛾subscript𝜓𝑛𝑦d(f^{n}y,f^{n}y^{\prime})\leq C_{2}\gamma^{\psi_{n}(y)}. As pointed out in [16], the stronger form (7.2) is satisfied in all known examples where the weaker form holds.

Condition (7.4) is not stated explicitly in [30] but is an automatic consequence of the set up therein provided f:XX:𝑓𝑋𝑋f:X\to X is injective. We provide the details in Proposition A.1. In the examples considered in this paper and in [26], the map f𝑓f is a first return map for a flow and hence is injective, so condition (7.4) is not very restrictive.

Condition (7.4) is also used in [26, Section 5.2] but is stated there in a slightly different form. In [26], the subspace X𝑋X is not needed (and hence not mentioned) and the stable and unstable disks Ws(y)superscript𝑊𝑠𝑦W^{s}(y), Wu(y)superscript𝑊𝑢𝑦W^{u}(y) are replaced by their intersections with Y𝑌Y. Hence the condition F(Wu(y)Yj)Wu(Fy)superscript𝑊𝑢𝐹𝑦𝐹superscript𝑊𝑢𝑦subscript𝑌𝑗F(W^{u}(y)\cap Y_{j})\supset W^{u}(Fy) for yYj𝑦subscript𝑌𝑗y\in Y_{j} in [26, Section 5.2] becomes F(Wu(y)Yj)Wu(Fy)Ysuperscript𝑊𝑢𝐹𝑦𝑌𝐹superscript𝑊𝑢𝑦subscript𝑌𝑗F(W^{u}(y)\cap Y_{j})\supset W^{u}(Fy)\cap Y for yYj𝑦subscript𝑌𝑗y\in Y_{j} in our present notation and hence holds by (7.4).

Proposition 7.3

d(fny,fny)C2C4(γψn(y)d(y,y)+γs(y,y)ψn(y))𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦subscript𝐶2subscript𝐶4superscript𝛾subscript𝜓𝑛𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦subscript𝜓𝑛𝑦d(f^{n}y,f^{n}y^{\prime})\leq C_{2}C_{4}(\gamma^{\psi_{n}(y)}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-\psi_{n}(y)}) for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, n0𝑛0n\geq 0.

Proof.

Let z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}). Note that s(z,y)=s(y,y)𝑠𝑧superscript𝑦𝑠𝑦superscript𝑦s(z,y^{\prime})=s(y,y^{\prime}) and ψn(z)=ψn(y)subscript𝜓𝑛𝑧subscript𝜓𝑛𝑦\psi_{n}(z)=\psi_{n}(y). Hence

d(fny,fny)d(fny,fnz)+d(fnz,fny)𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛𝑧𝑑superscript𝑓𝑛𝑧superscript𝑓𝑛superscript𝑦\displaystyle d(f^{n}y,f^{n}y^{\prime})\leq d(f^{n}y,f^{n}z)+d(f^{n}z,f^{n}y^{\prime}) C2(γψn(y)d(y,z)+γs(z,y)ψn(z))absentsubscript𝐶2superscript𝛾subscript𝜓𝑛𝑦𝑑𝑦𝑧superscript𝛾𝑠𝑧superscript𝑦subscript𝜓𝑛𝑧\displaystyle\leq C_{2}(\gamma^{\psi_{n}(y)}d(y,z)+\gamma^{s(z,y^{\prime})-\psi_{n}(z)})
C2C4(γψn(y)d(y,y)+γs(y,y)ψn(y)),absentsubscript𝐶2subscript𝐶4superscript𝛾subscript𝜓𝑛𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦subscript𝜓𝑛𝑦\displaystyle\leq C_{2}C_{4}(\gamma^{\psi_{n}(y)}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-\psi_{n}(y)}),

as required. ∎

7.2 Hölder flows and observables

Let Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M be a flow defined on a metric space (M,d)𝑀𝑑(M,d) with diamM1diam𝑀1\operatorname{diam}M\leq 1. Fix η(0,1]𝜂01\eta\in(0,1].

Given v:M:𝑣𝑀v:M\to{\mathbb{R}}, define |v|Cη=supxx|v(x)v(x)|/d(x,x)ηsubscript𝑣superscript𝐶𝜂subscriptsupremum𝑥superscript𝑥𝑣𝑥𝑣superscript𝑥𝑑superscript𝑥superscript𝑥𝜂{|v|}_{C^{\eta}}=\sup_{x\neq x^{\prime}}|v(x)-v(x^{\prime})|/d(x,x^{\prime})^{\eta} and vCη=|v|+|v|Cηsubscriptnorm𝑣superscript𝐶𝜂subscript𝑣subscript𝑣superscript𝐶𝜂{\|v\|}_{C^{\eta}}=|v|_{\infty}+{|v|}_{C^{\eta}}. Let Cη(M)={v:M:vCη<}superscript𝐶𝜂𝑀conditional-set𝑣:𝑀subscriptnorm𝑣superscript𝐶𝜂C^{\eta}(M)=\{v:M\to{\mathbb{R}}:{\|v\|}_{C^{\eta}}<\infty\}. Also, define |v|C0,η=supxM,t>0|v(Ttx)v(x)|/tηsubscript𝑣superscript𝐶0𝜂subscriptsupremumformulae-sequence𝑥𝑀𝑡0𝑣subscript𝑇𝑡𝑥𝑣𝑥superscript𝑡𝜂{|v|}_{C^{0,\eta}}=\sup_{x\in M,\,t>0}|v(T_{t}x)-v(x)|/t^{\eta} and let C0,η(M)={v:M:|v|+|v|C0,η<}superscript𝐶0𝜂𝑀conditional-set𝑣:𝑀subscript𝑣subscript𝑣superscript𝐶0𝜂C^{0,\eta}(M)=\{v:M\to{\mathbb{R}}:|v|_{\infty}+{|v|}_{C^{0,\eta}}<\infty\}. (Such observables are Hölder in the flow direction.)

We say that w:M:𝑤𝑀w:M\to{\mathbb{R}} is differentiable in the flow direction if the limit tw=limt0(wTtw)/tsubscript𝑡𝑤subscript𝑡0𝑤subscript𝑇𝑡𝑤𝑡\partial_{t}w=\lim_{t\to 0}(w\circ T_{t}-w)/t exists pointwise. Define wCη,m=j=0mtjwCηsubscriptnorm𝑤superscript𝐶𝜂𝑚superscriptsubscript𝑗0𝑚subscriptnormsuperscriptsubscript𝑡𝑗𝑤superscript𝐶𝜂{\|w\|}_{C^{\eta,m}}=\sum_{j=0}^{m}{\|\partial_{t}^{j}w\|}_{C^{\eta}} and let Cη,m(M)={w:M:wCη,m<}superscript𝐶𝜂𝑚𝑀conditional-set𝑤:𝑀subscriptnorm𝑤superscript𝐶𝜂𝑚C^{\eta,m}(M)=\{w:M\to{\mathbb{R}}:{\|w\|}_{C^{\eta,m}}<\infty\}.

Let XM𝑋𝑀X\subset M be a Borel subset and define Cη(X)superscript𝐶𝜂𝑋C^{\eta}(X) using the metric d𝑑d restricted to X𝑋X. We suppose that Th(x)xXsubscript𝑇𝑥𝑥𝑋T_{h(x)}x\in X for all xX𝑥𝑋x\in X, where h:X+:𝑋superscripth:X\to{\mathbb{R}}^{+} lies in Cη(X)superscript𝐶𝜂𝑋C^{\eta}(X) and infh>0infimum0\inf h>0. In addition, we suppose that for any D1>0subscript𝐷10D_{1}>0 there exists D2>0subscript𝐷20D_{2}>0 such that

d(Ttx,Ttx)D2d(x,x)ηfor all t[0,D1]x,xM.𝑑subscript𝑇𝑡𝑥subscript𝑇𝑡superscript𝑥subscript𝐷2𝑑superscript𝑥superscript𝑥𝜂for all t[0,D1]x,xM.d(T_{t}x,T_{t}x^{\prime})\leq D_{2}d(x,x^{\prime})^{\eta}\quad\text{for all $t\in[0,D_{1}]$, $x,x^{\prime}\in M$.} (7.5)

Define f:XX:𝑓𝑋𝑋f:X\to X by fx=Th(x)x𝑓𝑥subscript𝑇𝑥𝑥fx=T_{h(x)}x. We suppose that f𝑓f is a nonuniformly hyperbolic transformation as in Subsection 7.1, with induced map F=fτ:YY:𝐹superscript𝑓𝜏𝑌𝑌F=f^{\tau}:Y\to Y and so on.

Define h=j=01hfjsubscriptsuperscriptsubscript𝑗01superscript𝑓𝑗h_{\ell}=\sum_{j=0}^{\ell-1}h\circ f^{j}. We define the induced roof function

φ=hτ:Y+,φ(y)==0τ(y)1h(fy).:𝜑subscript𝜏formulae-sequence𝑌superscript𝜑𝑦superscriptsubscript0𝜏𝑦1superscript𝑓𝑦\textstyle\varphi=h_{\tau}:Y\to{\mathbb{R}}^{+},\qquad\varphi(y)=\sum_{\ell=0}^{\tau(y)-1}h(f^{\ell}y).

Note that hφ|h|τ𝜑subscript𝜏h\leq\varphi\leq|h|_{\infty}\tau so φL1(Y)𝜑superscript𝐿1𝑌\varphi\in L^{1}(Y) and infφ>0infimum𝜑0\inf\varphi>0. Define the suspension flow Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} as in (1.1).

To deduce rates of mixing for nonuniformly hyperbolic flows from the corresponding result for Gibbs-Markov flows, Theorem 6.4, we need to verify that

  • (i)

    Condition (6.1) holds.

  • (ii)

    Condition (H) from Section 6 holds.

  • (iii)

    Regular observables on M𝑀M lift to regular observables on Yφsuperscript𝑌𝜑Y^{\varphi}.

Ingredients (i) and (ii) guarantee that the suspension flow Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow and ingredient (iii) ensures that Theorem 6.4 applies to the appropriate observables on M𝑀M.

In the remainder of this subsection, we deal with ingredients (i) and (iii). First, we verify that φ𝜑\varphi satisfies condition (6.1). Let d1(y,y)=d(y,y)ηsubscript𝑑1𝑦superscript𝑦𝑑superscript𝑦superscript𝑦𝜂d_{1}(y,y^{\prime})=d(y,y^{\prime})^{\eta} and γ1=γηsubscript𝛾1superscript𝛾𝜂\gamma_{1}=\gamma^{\eta}.

Proposition 7.4

Let y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j} for some j1𝑗1j\geq 1 and let =0,,τ(y)10𝜏𝑦1\ell=0,\ldots,\tau(y)-1. Then

|h(y)h(y)|C2C4|h|η(d1(y,y)+γ1s(y,y)).subscript𝑦subscriptsuperscript𝑦subscript𝐶2subscript𝐶4subscript𝜂subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦|h_{\ell}(y)-h_{\ell}(y^{\prime})|\leq C_{2}C_{4}|h|_{\eta}\,\ell(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}).

Moreover,

|φ(y)φ(y)|2C22C4(infh)1|h|ηinfYjφγ1s(y,y)for all y,yY~jj1.𝜑𝑦𝜑superscript𝑦2superscriptsubscript𝐶22subscript𝐶4superscriptinfimum1subscript𝜂subscriptinfimumsubscript𝑌𝑗𝜑superscriptsubscript𝛾1𝑠𝑦superscript𝑦for all y,yY~jj1.|\varphi(y)-\varphi(y^{\prime})|\leq 2C_{2}^{2}C_{4}(\inf h)^{-1}|h|_{\eta}\,{\textstyle\inf_{Y_{j}}}\varphi\,\gamma_{1}^{s(y,y^{\prime})}\quad\text{for all $y,y^{\prime}\in{\widetilde{Y}}_{j}$, $j\geq 1$.}
Proof.

Note that ψ(y)=0subscript𝜓𝑦0\psi_{\ell}(y)=0, so by Proposition 7.3,

d(fy,fy)C2C4(d(y,y)+γs(y,y)).𝑑superscript𝑓𝑦superscript𝑓superscript𝑦subscript𝐶2subscript𝐶4𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦d(f^{\ell}y,f^{\ell}y^{\prime})\leq C_{2}C_{4}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}). (7.6)

Hence

|h(y)h(y)|subscript𝑦subscriptsuperscript𝑦\displaystyle|h_{\ell}(y)-h_{\ell}(y^{\prime})| j=01|h(fjy)h(fjy)|absentsuperscriptsubscript𝑗01superscript𝑓𝑗𝑦superscript𝑓𝑗superscript𝑦\displaystyle\leq\sum_{j=0}^{\ell-1}|h(f^{j}y)-h(f^{j}y^{\prime})|
|h|ηj=01d(fjy,fjy)ηC2C4|h|η(d1(y,y)+γ1s(y,y)),absentsubscript𝜂superscriptsubscript𝑗01𝑑superscriptsuperscript𝑓𝑗𝑦superscript𝑓𝑗superscript𝑦𝜂subscript𝐶2subscript𝐶4subscript𝜂subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦\displaystyle\leq|h|_{\eta}\sum_{j=0}^{\ell-1}d(f^{j}y,f^{j}y^{\prime})^{\eta}\leq C_{2}C_{4}|h|_{\eta}\,\ell(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}),

establishing the estimate for hsubscripth_{\ell}. Also, τ(y)(infh)1inf1Yjφ𝜏𝑦superscriptinfimum1infimumsubscript1subscript𝑌𝑗𝜑\tau(y)\leq(\inf h)^{-1}\inf 1_{Y_{j}}\varphi, so taking =τ(y)𝜏𝑦\ell=\tau(y) and using (7.3) with n=0𝑛0n=0, we obtain the estimate for φ𝜑\varphi. ∎

Next we deal with ingredient (iii) assuming (ii). Define π:YφM:𝜋superscript𝑌𝜑𝑀\pi:Y^{\varphi}\to M as π(y,u)=Tuy𝜋𝑦𝑢subscript𝑇𝑢𝑦\pi(y,u)=T_{u}y.

Proposition 7.5

Suppose that the function χ:Y:𝜒𝑌\chi:Y\to{\mathbb{R}} satisfies condition (H).

Then observables vCη(M)C0,η(M)𝑣superscript𝐶𝜂𝑀superscript𝐶0𝜂𝑀v\in C^{\eta}(M)\cap C^{0,\eta}(M) lift to observables v~=vπ:Yφ:~𝑣𝑣𝜋superscript𝑌𝜑\tilde{v}=v\circ\pi:Y^{\varphi}\to{\mathbb{R}} that lie in γ2,η(Yφ)subscriptsuperscriptsubscript𝛾2𝜂superscript𝑌𝜑{\mathcal{H}}^{*}_{\gamma_{2},\eta}(Y^{\varphi}) where γ2=γη2subscript𝛾2superscript𝛾superscript𝜂2\gamma_{2}=\gamma^{\eta^{2}} and the metric d𝑑d on Y𝑌Y is replaced by the metric d2(y,y)=d(y,y)η2subscript𝑑2𝑦superscript𝑦𝑑superscript𝑦superscript𝑦superscript𝜂2d_{2}(y,y^{\prime})=d(y,y^{\prime})^{\eta^{2}}.

For m1𝑚1m\geq 1, observables wCη,m(M)𝑤superscript𝐶𝜂𝑚𝑀w\in C^{\eta,m}(M) lift to observables w~=wπγ2,0,m(Yφ)~𝑤𝑤𝜋subscriptsuperscriptsubscript𝛾20𝑚superscript𝑌𝜑\tilde{w}=w\circ\pi\in{\mathcal{H}}^{*}_{\gamma_{2},0,m}(Y^{\varphi}).

Moreover, there is a constant C>0𝐶0C>0 such that v~γ2,ηC(vCη+vC0,η)subscriptsuperscriptnorm~𝑣subscript𝛾2𝜂𝐶subscriptnorm𝑣superscript𝐶𝜂subscriptnorm𝑣superscript𝐶0𝜂\|\tilde{v}\|^{*}_{\gamma_{2},\eta}\leq C({\|v\|}_{C^{\eta}}+{\|v\|}_{C^{0,\eta}}) and w~γ2,0,mCwCη,msubscriptsuperscriptnorm~𝑤subscript𝛾20𝑚𝐶subscriptnorm𝑤superscript𝐶𝜂𝑚\|\tilde{w}\|^{*}_{\gamma_{2},0,m}\leq C{\|w\|}_{C^{\eta,m}}.

Proof.

Let σ=2|χ|𝜎2subscript𝜒\sigma=2|\chi|_{\infty}. We show that v~Fσγ2,ηvCη+vC0,ηmuch-less-thansubscriptnorm~𝑣subscript𝐹𝜎subscript𝛾2𝜂subscriptnorm𝑣superscript𝐶𝜂subscriptnorm𝑣superscript𝐶0𝜂\|\tilde{v}\circ F_{\sigma}\|_{\gamma_{2},\eta}\ll{\|v\|}_{C^{\eta}}+{\|v\|}_{C^{0,\eta}}. The same calculation with σ=0𝜎0\sigma=0 shows that v~γ2,ηvCη+vC0,ηmuch-less-thansubscriptnorm~𝑣subscript𝛾2𝜂subscriptnorm𝑣superscript𝐶𝜂subscriptnorm𝑣superscript𝐶0𝜂\|\tilde{v}\|_{\gamma_{2},\eta}\ll{\|v\|}_{C^{\eta}}+{\|v\|}_{C^{0,\eta}}, so v~γ2,ηvCη+vC0,ηmuch-less-thansubscriptsuperscriptnorm~𝑣subscript𝛾2𝜂subscriptnorm𝑣superscript𝐶𝜂subscriptnorm𝑣superscript𝐶0𝜂\|\tilde{v}\|^{*}_{\gamma_{2},\eta}\ll{\|v\|}_{C^{\eta}}+{\|v\|}_{C^{0,\eta}}, We take D1=|h|+2|χ|subscript𝐷1subscript2subscript𝜒D_{1}=|h|_{\infty}+2|\chi|_{\infty} with corresponding value of D2subscript𝐷2D_{2} in (7.5)

Let (y,u),(y,u)Yφ𝑦𝑢superscript𝑦𝑢superscript𝑌𝜑(y,u),\,(y^{\prime},u)\in Y^{\varphi} with y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j} for some j1𝑗1j\geq 1. There exists ,{0,,τ(y)1}superscript0𝜏𝑦1\ell,\ell^{\prime}\in\{0,\dots,\tau(y)-1\} such that

u[h(y),h+1(y)][h(y),h+1(y)].𝑢subscript𝑦subscript1𝑦subscriptsuperscriptsuperscript𝑦subscriptsuperscript1superscript𝑦u\in[h_{\ell}(y),h_{\ell+1}(y)]\cap[h_{\ell^{\prime}}(y^{\prime}),h_{\ell^{\prime}+1}(y^{\prime})].

Suppose without loss that superscript\ell\leq\ell^{\prime}. Then

u=h(y)+r=h(y)+r,𝑢subscript𝑦𝑟subscriptsuperscript𝑦superscript𝑟u=h_{\ell}(y)+r=h_{\ell}(y^{\prime})+r^{\prime},

where r[0,|h|]𝑟0subscriptr\in[0,|h|_{\infty}] and r=uh(y)uh(y)0superscript𝑟𝑢subscriptsuperscript𝑦𝑢subscriptsuperscriptsuperscript𝑦0r^{\prime}=u-h_{\ell}(y^{\prime})\geq u-h_{\ell^{\prime}}(y^{\prime})\geq 0. Note that Tuy=TrTh(y)y=Trfysubscript𝑇𝑢𝑦subscript𝑇𝑟subscript𝑇subscript𝑦𝑦subscript𝑇𝑟superscript𝑓𝑦T_{u}y=T_{r}T_{h_{\ell}(y)}y=T_{r}f^{\ell}y. Hence v~(y,u)=v(Trfy)~𝑣𝑦𝑢𝑣subscript𝑇𝑟superscript𝑓𝑦\tilde{v}(y,u)=v(T_{r}f^{\ell}y) and so v~Fσ(y,u)=v(Tσ+rfy)~𝑣subscript𝐹𝜎𝑦𝑢𝑣subscript𝑇𝜎𝑟superscript𝑓𝑦\tilde{v}\circ F_{\sigma}(y,u)=v(T_{\sigma+r}f^{\ell}y). Similarly, Tuy=Trfysubscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑟superscript𝑓superscript𝑦T_{u}y^{\prime}=T_{r^{\prime}}f^{\ell}y^{\prime} and v~Fσ(y,u)=v(Tσ+rfy)~𝑣subscript𝐹𝜎superscript𝑦𝑢𝑣subscript𝑇𝜎superscript𝑟superscript𝑓superscript𝑦\tilde{v}\circ F_{\sigma}(y^{\prime},u)=v(T_{\sigma+r^{\prime}}f^{\ell}y^{\prime}). Also, σ+r[0,D1]𝜎𝑟0subscript𝐷1\sigma+r\in[0,D_{1}]. By (7.5) and (7.6),

|v(Tσ+rfy)v(Tσ+rfy)|𝑣subscript𝑇𝜎𝑟superscript𝑓𝑦𝑣subscript𝑇𝜎𝑟superscript𝑓superscript𝑦\displaystyle|v(T_{\sigma+r}f^{\ell}y)-v(T_{\sigma+r}f^{\ell}y^{\prime})| |v|Cηd(Tσ+rfy,Tσ+rfy)ηD2η|v|Cηd(fy,fy)η2absentsubscript𝑣superscript𝐶𝜂𝑑superscriptsubscript𝑇𝜎𝑟superscript𝑓𝑦subscript𝑇𝜎𝑟superscript𝑓superscript𝑦𝜂superscriptsubscript𝐷2𝜂subscript𝑣superscript𝐶𝜂𝑑superscriptsuperscript𝑓𝑦superscript𝑓superscript𝑦superscript𝜂2\displaystyle\leq{|v|}_{C^{\eta}}d(T_{\sigma+r}f^{\ell}y,T_{\sigma+r}f^{\ell}y^{\prime})^{\eta}\leq D_{2}^{\eta}{|v|}_{C^{\eta}}d(f^{\ell}y,f^{\ell}y^{\prime})^{\eta^{2}}
|v|Cη(d2(y,y)+γ2s(y,y)).much-less-thanabsentsubscript𝑣superscript𝐶𝜂subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll{|v|}_{C^{\eta}}(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

Since uh(y)infh𝑢subscriptsuperscript𝑦infimumu\geq h_{\ell}(y^{\prime})\geq\ell\inf h, it follows from Proposition 7.4 that

|v(Tσ+rfy)\displaystyle|v(T_{\sigma+r}f^{\ell}y^{\prime})- v(Tσ+rfy)||v|C0,η|rr|η=|v|C0,η|h(y)h(y)|η\displaystyle v(T_{\sigma+r^{\prime}}f^{\ell}y^{\prime})|\leq{|v|}_{C^{0,\eta}}|r-r^{\prime}|^{\eta}={|v|}_{C^{0,\eta}}|h_{\ell}(y)-h_{\ell}(y^{\prime})|^{\eta}
|v|C0,η(d2(y,y)η+γ2s(y,y))(infh)1|v|C0,ηu(d2(y,y)η+γ2s(y,y)).much-less-thanabsentsubscript𝑣superscript𝐶0𝜂subscript𝑑2superscript𝑦superscript𝑦𝜂superscriptsubscript𝛾2𝑠𝑦superscript𝑦superscriptinfimum1subscript𝑣superscript𝐶0𝜂𝑢subscript𝑑2superscript𝑦superscript𝑦𝜂superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll{|v|}_{C^{0,\eta}}\,\ell(d_{2}(y,y^{\prime})^{\eta}+\gamma_{2}^{s(y,y^{\prime})})\leq(\inf h)^{-1}{|v|}_{C^{0,\eta}}\,u(d_{2}(y,y^{\prime})^{\eta}+\gamma_{2}^{s(y,y^{\prime})}).

Hence

|v~Fσ(y,u)v~Fσ(y,u)|~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢\displaystyle|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)| =|v(Tσ+rfy)v(Tσ+rfy)|absent𝑣subscript𝑇𝜎𝑟superscript𝑓𝑦𝑣subscript𝑇𝜎superscript𝑟superscript𝑓superscript𝑦\displaystyle=|v(T_{\sigma+r}f^{\ell}y)-v(T_{\sigma+r^{\prime}}f^{\ell}y^{\prime})|
(|v|Cη+|v|C0,η)(u+1)(d2(y,y)+γ2s(y,y))much-less-thanabsentsubscript𝑣superscript𝐶𝜂subscript𝑣superscript𝐶0𝜂𝑢1subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll({|v|}_{C^{\eta}}+{|v|}_{C^{0,\eta}})(u+1)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})})

whenever s(y,y)1𝑠𝑦superscript𝑦1s(y,y^{\prime})\geq 1. For s(y,y)=0𝑠𝑦superscript𝑦0s(y,y^{\prime})=0, we have the estimate |v~Fσ(y,u)v~Fσ(y,u)|2|v|=2|v|γ2s(y,y)|v|φ(y)(d2(y,y)+γ2s(y,y))~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢2subscript𝑣2subscript𝑣superscriptsubscript𝛾2𝑠𝑦superscript𝑦much-less-thansubscript𝑣𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)|\leq 2|v|_{\infty}=2|v|_{\infty}\gamma_{2}^{s(y,y^{\prime})}\ll|v|_{\infty}\,\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}), so in all cases we obtain

|v~Fσ(y,u)v~Fσ(y,u)|~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢\displaystyle|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)| (vCη+|v|C0,η)(u+1)(d2(y,y)+γ2s(y,y))much-less-thanabsentsubscriptnorm𝑣superscript𝐶𝜂subscript𝑣superscript𝐶0𝜂𝑢1subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll({\|v\|}_{C^{\eta}}+{|v|}_{C^{0,\eta}})(u+1)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})})
2(vCη+|v|C0,η)φ(y)(d2(y,y)+γ2s(y,y)).absent2subscriptnorm𝑣superscript𝐶𝜂subscript𝑣superscript𝐶0𝜂𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\leq 2({\|v\|}_{C^{\eta}}+{|v|}_{C^{0,\eta}})\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

Also,

|v~Fσ(y,u)v~Fσ(y,u)|=|v(Tσ+uy)v(Tσ+uy)|v|C0,η|uu|η,|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y,u^{\prime})|=|v(T_{\sigma+u}y)-v(T_{\sigma+u^{\prime}}y)\leq{|v|}_{C^{0,\eta}}|u-u^{\prime}|^{\eta},

so v~Fσγ2.η|v|C0+|v|C0,ηmuch-less-thansubscriptnorm~𝑣subscript𝐹𝜎formulae-sequencesubscript𝛾2𝜂subscript𝑣superscript𝐶0subscript𝑣superscript𝐶0𝜂\|\tilde{v}\circ F_{\sigma}\|_{\gamma_{2}.\eta}\ll{|v|}_{C^{0}}+{|v|}_{C^{0,\eta}} as required. ∎

7.3 Dynamically Hölder flows and observables

The Hölder assumptions in Subsection 7.2 can be replaced by dynamically Hölder as follows. We continue to assume that infh>0infimum0\inf h>0.

Definition 7.6

The roof function hh, the flow Ttsubscript𝑇𝑡T_{t} and the observable v𝑣v are dynamically Hölder if vC0,η(M)𝑣superscript𝐶0𝜂𝑀v\in C^{0,\eta}(M) for some η(0,1]𝜂01\eta\in(0,1] and there is a constant C1𝐶1C\geq 1 such that for all y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j}, j1𝑗1j\geq 1,

  • (a)

    |h(fy)h(fy)|C(d(y,y)η+γs(y,y))superscript𝑓𝑦superscript𝑓superscript𝑦𝐶𝑑superscript𝑦superscript𝑦𝜂superscript𝛾𝑠𝑦superscript𝑦|h(f^{\ell}y)-h(f^{\ell}y^{\prime})|\leq C(d(y,y^{\prime})^{\eta}+\gamma^{s(y,y^{\prime})}) for all 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1.

  • (b)

    For every u[0,φ(y)][0,φ(y)]𝑢0𝜑𝑦0𝜑superscript𝑦u\in[0,\varphi(y)]\cap[0,\varphi(y^{\prime})], there exist t,t𝑡superscript𝑡t,t^{\prime}\in{\mathbb{R}} such that |tt|C(u+1)(d(y,y)η+γs(y,y))𝑡superscript𝑡𝐶𝑢1𝑑superscript𝑦superscript𝑦𝜂superscript𝛾𝑠𝑦superscript𝑦|t-t^{\prime}|\leq C{(u+1)}(d(y,y^{\prime})^{\eta}+\gamma^{s(y,y^{\prime})}), and setting z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}),

    max{|v(Tuy)v(Ttz)|,|v(Tuy)v(Ttz)|}C(u+1)(d(y,y)η+γs(y,y)).𝑣subscript𝑇𝑢𝑦𝑣subscript𝑇𝑡𝑧𝑣subscript𝑇𝑢superscript𝑦𝑣subscript𝑇superscript𝑡𝑧𝐶𝑢1𝑑superscript𝑦superscript𝑦𝜂superscript𝛾𝑠𝑦superscript𝑦\max\big{\{}|v(T_{u}y)-v(T_{t}z)|\,,\,|v(T_{u}y^{\prime})-v(T_{t^{\prime}}z)|\big{\}}\leq C(u+1)(d(y,y^{\prime})^{\eta}+\gamma^{s(y,y^{\prime})}).

Also, we replace the assumption wCη,m(M)𝑤superscript𝐶𝜂𝑚𝑀w\in C^{\eta,m}(M) by the condition that tkwsuperscriptsubscript𝑡𝑘𝑤\partial_{t}^{k}w lies in C0,η(M)superscript𝐶0𝜂𝑀C^{0,\eta}(M) and satisfies (b) for all k=0,,m𝑘0𝑚k=0,\dots,m.

Remark 7.7

In the proof of Proposition 7.5, we showed that |v(Tuy)v(Tuy)|=|v~(y,u)v~(y,u)|(u+1)(d(y,y)η+γs(y,y))𝑣subscript𝑇𝑢𝑦𝑣subscript𝑇𝑢superscript𝑦~𝑣𝑦𝑢~𝑣superscript𝑦𝑢much-less-than𝑢1𝑑superscript𝑦superscript𝑦𝜂superscript𝛾𝑠𝑦superscript𝑦|v(T_{u}y)-v(T_{u}y^{\prime})|=|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|\ll(u+1)(d(y,y^{\prime})^{\eta}+\gamma^{s(y,y^{\prime})}) (for modified d𝑑d and γ𝛾\gamma) under the old hypotheses. Hence, taking t=t=u𝑡superscript𝑡𝑢t=t^{\prime}=u, we see that Definition 7.6 is indeed a relaxed version of the conditions in Subsection 7.2.

It is easily verified that condition (6.1) remains valid under the more relaxed assumption on hh in Definition 7.6(a). Also, it follows as in the proof of Proposition 7.5 that |v~(y,u)v~(y,u)||v|C0,η|uu|η~𝑣𝑦𝑢~𝑣𝑦superscript𝑢subscript𝑣superscript𝐶0𝜂superscript𝑢superscript𝑢𝜂|\tilde{v}(y,u)-\tilde{v}(y,u^{\prime})|\leq{|v|}_{C^{0,\eta}}|u-u^{\prime}|^{\eta}.

Next we estimate |v~(y,u)v~(y,u)|~𝑣𝑦𝑢~𝑣superscript𝑦𝑢|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)| and |v~Fσ(y,u)v~Fσ(y,u)|~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)| for (y,u),(y,u)Yφ𝑦𝑢superscript𝑦𝑢superscript𝑌𝜑(y,u),\,(y^{\prime},u)\in Y^{\varphi}, where σ=2|χ|𝜎2subscript𝜒\sigma=2|\chi|_{\infty}. If s(y,y)=0𝑠𝑦superscript𝑦0s(y,y^{\prime})=0, then |v~(y,u)v~(y,u)|,|v~Fσ(y,u)v~Fσ(y,u)||v|φ(y)(d2(y,y)+γ2s(y,y))much-less-than~𝑣𝑦𝑢~𝑣superscript𝑦𝑢~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢subscript𝑣𝜑𝑦subscript𝑑2𝑦𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|,\,|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)|\ll|v|_{\infty}\,\varphi(y)(d_{2}(y,y)+\gamma_{2}^{s(y,y^{\prime})}) as in the proof of Proposition 7.5. Hence we can suppose that y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j} for some j1𝑗1j\geq 1. Set z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}) and choose t,t𝑡superscript𝑡t,t^{\prime} as in Definition 7.6(b). Then

|v~(y,u)\displaystyle|\tilde{v}(y,u)- v~(y,u)|=|v(Tuy)v(Tuy)|\displaystyle\tilde{v}(y^{\prime},u)|=|v(T_{u}y)-v(T_{u}y^{\prime})|
|v(Tuy)v(Ttz)|+|v(Ttz)v(Tuy)|+|v(Ttz)v(Ttz)|absent𝑣subscript𝑇𝑢𝑦𝑣subscript𝑇𝑡𝑧𝑣subscript𝑇superscript𝑡𝑧𝑣subscript𝑇𝑢superscript𝑦𝑣subscript𝑇𝑡𝑧𝑣subscript𝑇superscript𝑡𝑧\displaystyle\leq|v(T_{u}y)-v(T_{t}z)|+|v(T_{t^{\prime}}z)-v(T_{u}y^{\prime})|+|v(T_{t}z)-v(T_{t^{\prime}}z)|
4Cφ(y)(d(y,y)η+γs(y,y))+|v|C0,η|tt|ηφ(y)(d2(y,y)+γ2s(y,y)).absent4𝐶𝜑𝑦𝑑superscript𝑦superscript𝑦𝜂superscript𝛾𝑠𝑦superscript𝑦subscript𝑣superscript𝐶0𝜂superscript𝑡superscript𝑡𝜂much-less-than𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\leq 4C\varphi(y)(d(y,y^{\prime})^{\eta}+\gamma^{s(y,y^{\prime})})+{|v|}_{C^{0,\eta}}|t-t^{\prime}|^{\eta}\ll\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

Hence |v~(y,u)v~(y,u)|φ(y)(d2(y,y)η+γ2s(y,y))much-less-than~𝑣𝑦𝑢~𝑣superscript𝑦𝑢𝜑𝑦subscript𝑑2superscript𝑦superscript𝑦𝜂superscriptsubscript𝛾2𝑠𝑦superscript𝑦|\tilde{v}(y,u)-\tilde{v}(y^{\prime},u)|\ll\varphi(y)(d_{2}(y,y^{\prime})^{\eta}+\gamma_{2}^{s(y,y^{\prime})}) for all (y,u),(y,u)Yφ𝑦𝑢superscript𝑦𝑢superscript𝑌𝜑(y,u),\,(y^{\prime},u)\in Y^{\varphi}, and so v~γ2,η(Yφ)~𝑣subscriptsubscript𝛾2𝜂superscript𝑌𝜑\tilde{v}\in{\mathcal{H}}_{\gamma_{2},\eta}(Y^{\varphi}).

To proceed, we recall that z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}), so Fz=Ws(Fy)Wu(Fy)𝐹𝑧superscript𝑊𝑠𝐹𝑦superscript𝑊𝑢𝐹superscript𝑦Fz=W^{s}(Fy)\cap W^{u}(Fy^{\prime}) by (7.4). Hence

d(Fy,Fy)d(Fy,Fz)+d(Fz,Fy)d(y,z)+γs(y,y)C4d(y,y)+γs(y,y).𝑑𝐹𝑦𝐹superscript𝑦𝑑𝐹𝑦𝐹𝑧𝑑𝐹𝑧𝐹superscript𝑦much-less-than𝑑𝑦𝑧superscript𝛾𝑠𝑦superscript𝑦subscript𝐶4𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦d(Fy,Fy^{\prime})\leq d(Fy,Fz)+d(Fz,Fy^{\prime})\ll d(y,z)+\gamma^{s(y,y^{\prime})}\leq C_{4}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}. (7.7)

To control v~Fσ(y,u)v~Fσ(y,u)~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u), we assume without loss that φ(y)φ(y)𝜑𝑦𝜑superscript𝑦\varphi(y)\geq\varphi(y^{\prime}), and distinguish three cases.

If u+σ<φ(y)𝑢𝜎𝜑superscript𝑦u+\sigma<\varphi(y^{\prime}), we argue as in the bound for v~(y,u)v~(y,u)~𝑣𝑦𝑢~𝑣superscript𝑦𝑢\tilde{v}(y,u)-\tilde{v}(y^{\prime},u).

If u+σφ(y)𝑢𝜎𝜑𝑦u+\sigma\geq\varphi(y), then there exists 0u¯σ0¯𝑢𝜎0\leq\bar{u}\leq\sigma and u¯u¯superscript¯𝑢¯𝑢\bar{u}^{\prime}\geq\bar{u} such that Tu+σy=Tu¯Fysubscript𝑇𝑢𝜎𝑦subscript𝑇¯𝑢𝐹𝑦T_{u+\sigma}y=T_{\bar{u}}Fy and Tu+σy=Tu¯Fysubscript𝑇𝑢𝜎superscript𝑦subscript𝑇superscript¯𝑢𝐹superscript𝑦T_{u+\sigma}y^{\prime}=T_{\bar{u}^{\prime}}Fy^{\prime}. By Corollary 6.2 and (7.7),

|u¯u¯|=|φ(y)φ(y)|φ(y)(d(y,y)+γs(y,y))¯𝑢superscript¯𝑢𝜑𝑦𝜑superscript𝑦much-less-than𝜑𝑦𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦|\bar{u}-\bar{u}^{\prime}|=|\varphi(y)-\varphi(y^{\prime})|\ll\varphi(y)(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})})

and so

|v(Tu¯Fy)v(Tu¯Fy)|φ(y)(d2(y,y)+γ2s(y,y)).much-less-than𝑣subscript𝑇¯𝑢𝐹superscript𝑦𝑣subscript𝑇superscript¯𝑢𝐹superscript𝑦𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦|v(T_{\bar{u}}Fy^{\prime})-v(T_{\bar{u}^{\prime}}Fy^{\prime})|\ll\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

On the other hand, choosing t¯¯𝑡\bar{t} and t¯superscript¯𝑡\bar{t}^{\prime} for u¯¯𝑢\bar{u} as in Definition 7.6(b), we get

|v(Tu¯Fy)\displaystyle|v(T_{\bar{u}}Fy) v(Tu¯Fy)|\displaystyle-v(T_{\bar{u}}Fy^{\prime})|
|v(Tu¯Fy)v(Tt¯Fz)|+|v(Tt¯Fz)v(Tu¯Fy)|+|v(Tt¯Fz)v(Tt¯Fz)|absent𝑣subscript𝑇¯𝑢𝐹𝑦𝑣subscript𝑇¯𝑡𝐹𝑧𝑣subscript𝑇superscript¯𝑡𝐹𝑧𝑣subscript𝑇¯𝑢𝐹superscript𝑦𝑣subscript𝑇¯𝑡𝐹𝑧𝑣subscript𝑇superscript¯𝑡𝐹𝑧\displaystyle\leq|v(T_{\bar{u}}Fy)-v(T_{\bar{t}}Fz)|+|v(T_{\bar{t}^{\prime}}Fz)-v(T_{\bar{u}}Fy^{\prime})|+|v(T_{\bar{t}}Fz)-v(T_{\bar{t}^{\prime}}Fz)|
2C(u¯+1)(d(Fy,Fy)η+γs(Fy,Fy))+|v|C0,η|t¯t¯|ηd2(y,y)+γ2s(y,y)absent2𝐶¯𝑢1𝑑superscript𝐹𝑦𝐹superscript𝑦𝜂superscript𝛾𝑠𝐹𝑦𝐹superscript𝑦subscript𝑣superscript𝐶0𝜂superscript¯𝑡superscript¯𝑡𝜂much-less-thansubscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\leq 2C(\bar{u}+1)(d(Fy,Fy^{\prime})^{\eta}+\gamma^{s(Fy,Fy^{\prime})})+{|v|}_{C^{0,\eta}}|\bar{t}-\bar{t}^{\prime}|^{\eta}\ll d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}

where we have used (7.7) and u¯σ¯𝑢𝜎\bar{u}\leq\sigma. Hence

|v~Fσ(y,u)v~Fσ(y,u)|~𝑣subscript𝐹𝜎𝑦𝑢~𝑣subscript𝐹𝜎superscript𝑦𝑢\displaystyle|\tilde{v}\circ F_{\sigma}(y,u)-\tilde{v}\circ F_{\sigma}(y^{\prime},u)| |v(Tu¯Fy)v(Tu¯Fy)|+|v(Tu¯Fy)v(Tu¯Fy)|absent𝑣subscript𝑇¯𝑢𝐹𝑦𝑣subscript𝑇¯𝑢𝐹superscript𝑦𝑣subscript𝑇¯𝑢𝐹superscript𝑦𝑣subscript𝑇superscript¯𝑢𝐹superscript𝑦\displaystyle\leq|v(T_{\bar{u}}Fy)-v(T_{\bar{u}}Fy^{\prime})|+|v(T_{\bar{u}}Fy^{\prime})-v(T_{\bar{u}^{\prime}}Fy^{\prime})|
φ(y)(d2(y,y)+γ2s(y,y)).much-less-thanabsent𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

Finally, if φ(y)u+σ<φ(y)𝜑superscript𝑦𝑢𝜎𝜑𝑦\varphi(y^{\prime})\leq u+\sigma<\varphi(y), there exist 0<u1,u2φ(y)φ(y)formulae-sequence0subscript𝑢1subscript𝑢2𝜑𝑦𝜑superscript𝑦0<u_{1},\,u_{2}\leq\varphi(y)-\varphi(y^{\prime}) such that Fy=Tu1Tu+σy𝐹𝑦subscript𝑇subscript𝑢1subscript𝑇𝑢𝜎𝑦Fy=T_{u_{1}}T_{u+\sigma}y and Tu+σy=Tu2Fysubscript𝑇𝑢𝜎superscript𝑦subscript𝑇subscript𝑢2𝐹superscript𝑦T_{u+\sigma}y^{\prime}=T_{u_{2}}Fy^{\prime}. Using again Corollary 6.2 and (7.7),

|v~Fσ(y,u)\displaystyle|\tilde{v}\circ F_{\sigma}(y,u)- v~Fσ(y,u)|=|v(Tu+σy)v(Tu+σy)|\displaystyle\tilde{v}\circ F_{\sigma}(y^{\prime},u)|=|v(T_{u+\sigma}y)-v(T_{u+\sigma}y^{\prime})|
|v(Tu+σy)v(Fy)|+|v(Fy)v(Fy)|+|v(Fy)v(Tu+σy)|absent𝑣subscript𝑇𝑢𝜎𝑦𝑣𝐹𝑦𝑣𝐹𝑦𝑣𝐹superscript𝑦𝑣𝐹superscript𝑦𝑣subscript𝑇𝑢𝜎superscript𝑦\displaystyle\leq|v(T_{u+\sigma}y)-v(Fy)|+|v(Fy)-v(Fy^{\prime})|+|v(Fy^{\prime})-v(T_{u+\sigma}y^{\prime})|
=|v(Tu+σy)v(Tu1+u+σy)|+|v(Fy)v(Fy)|+|v(Fy)v(Tu2Fy)|absent𝑣subscript𝑇𝑢𝜎𝑦𝑣subscript𝑇subscript𝑢1𝑢𝜎𝑦𝑣𝐹𝑦𝑣𝐹superscript𝑦𝑣𝐹superscript𝑦𝑣subscript𝑇subscript𝑢2𝐹superscript𝑦\displaystyle=|v(T_{u+\sigma}y)-v(T_{u_{1}+u+\sigma}y)|+|v(Fy)-v(Fy^{\prime})|+|v(Fy^{\prime})-v(T_{u_{2}}Fy^{\prime})|
φ(y)(d2(y,y)+γ2s(y,y)).much-less-thanabsent𝜑𝑦subscript𝑑2𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\ll\varphi(y)(d_{2}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

This completes the verification that v~γ2,η(Yφ)~𝑣subscriptsuperscriptsubscript𝛾2𝜂superscript𝑌𝜑\tilde{v}\in{\mathcal{H}}^{*}_{\gamma_{2},\eta}(Y^{\varphi}). A similar argument shows that w~γ2,0,m(Yφ)~𝑤subscriptsuperscriptsubscript𝛾20𝑚superscript𝑌𝜑\tilde{w}\in{\mathcal{H}}^{*}_{\gamma_{2},0,m}(Y^{\varphi}), completing the verification that Proposition 7.5 holds under the modified assumptions.

8 Condition (H) for nonuniformly hyperbolic flows

In this section, we consider various classes of nonuniformly hyperbolic flows for which condition (H) in Section 6 can be satisfied. We are then able to apply Theorem 6.4 to obtain results that superpolynomial and polynomial mixing applies to such flows as follows:

Corollary 8.1

Let Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M be a nonuniformly hyperbolic flow as in Section 7.2 and assume that condition (H) is satisfied. Then

(a) Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow.

(b) Suppose that μ(φ>t)=O(tβ)𝜇𝜑𝑡𝑂superscript𝑡𝛽\mu(\varphi>t)=O(t^{-\beta}) for some β>1𝛽1\beta>1 and assume absence of approximate eigenfunctions for Ftsubscript𝐹𝑡F_{t}. Then there exists m1𝑚1m\geq 1 and C>0𝐶0C>0 such that

|ρv,w(t)|C(vCη+vC0,η)wCη,mt(β1),subscript𝜌𝑣𝑤𝑡𝐶subscriptnorm𝑣superscript𝐶𝜂subscriptnorm𝑣superscript𝐶0𝜂subscriptnorm𝑤superscript𝐶𝜂𝑚superscript𝑡𝛽1|\rho_{v,w}(t)|\leq C({\|v\|}_{C^{\eta}}+{\|v\|}_{C^{0,\eta}}){\|w\|}_{C^{\eta,m}}\,t^{-(\beta-1)},

for all vCη(M)C0,η(M)𝑣superscript𝐶𝜂𝑀superscript𝐶0𝜂𝑀v\in C^{\eta}(M)\cap C^{0,\eta}(M), wCη,m(M)𝑤superscript𝐶𝜂𝑚𝑀w\in C^{\eta,m}(M), t>1𝑡1t>1.

Proof.

Part (a) follows from the discussion in Section 7.2 (so ingredient (i) is automatic and ingredient (ii) is now assumed).

As described in Section 6.1, there is a measure-preserving conjugacy from Ftsubscript𝐹𝑡F_{t} to Ttsubscript𝑇𝑡T_{t}, so part (b) is immediate from Theorem 6.4 combined with Proposition 7.5. ∎

The analogous result holds for nonuniformly hyperbolic flows and observables satisfying the dynamically Hölder conditions in Section 7.3.

We verify condition (H) for three classes of flows. In Subsection 8.1, we consider roof functions with bounded Hölder constants. In Subsection 8.2, we consider flows for which there is exponential contraction along stable leaves. In Subsection 8.3, we consider flows with an invariant Hölder stable foliation. These correspond to the situations mentioned in [26, Section 4.2].

Also, in Subsection 8.4, we briefly review the temporal distance function and a criterion for absence of approximate eigenfunctions.

8.1 Roof functions with bounded Hölder constants

We assume a “bounded Hölder constants” condition on φ𝜑\varphi, namely that for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y,

|φ(y)φ(y)|𝜑𝑦𝜑superscript𝑦\displaystyle|\varphi(y)-\varphi(y^{\prime})| C1d(y,y)absentsubscript𝐶1𝑑𝑦superscript𝑦\displaystyle\leq C_{1}d(y,y^{\prime}) for all yWs(y)superscript𝑦superscript𝑊𝑠𝑦y^{\prime}\in W^{s}(y), (8.1)
|φ(y)φ(y)|𝜑𝑦𝜑superscript𝑦\displaystyle|\varphi(y)-\varphi(y^{\prime})| C1γs(y,y)absentsubscript𝐶1superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq C_{1}\gamma^{s(y,y^{\prime})} for all yWu(y)superscript𝑦superscript𝑊𝑢𝑦y^{\prime}\in W^{u}(y), s(y,y)1𝑠𝑦superscript𝑦1s(y,y^{\prime})\geq 1. (8.2)

This leads directly to an enhanced version of (6.1):

Proposition 8.2

|φ(y)φ(y)|C1C4(d(y,y)+γs(y,y))𝜑𝑦𝜑superscript𝑦subscript𝐶1subscript𝐶4𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦|\varphi(y)-\varphi(y^{\prime})|\leq C_{1}C_{4}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}) for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, s(y,y)1𝑠𝑦superscript𝑦1s(y,y^{\prime})\geq 1.

Proof.

Let z=Ws(y)Ws(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑠superscript𝑦z=W^{s}(y)\cap W^{s}(y^{\prime}). Then

|φ(y)φ(y)|𝜑𝑦𝜑superscript𝑦\displaystyle|\varphi(y)-\varphi(y^{\prime})| |φ(y)φ(z)|+|φ(z)φ(y)|absent𝜑𝑦𝜑𝑧𝜑𝑧𝜑superscript𝑦\displaystyle\leq|\varphi(y)-\varphi(z)|+|\varphi(z)-\varphi(y^{\prime})|
C1(d(y,z)+γs(z,y))C1C4(d(y,y)+γs(y,y)),absentsubscript𝐶1𝑑𝑦𝑧superscript𝛾𝑠𝑧superscript𝑦subscript𝐶1subscript𝐶4𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq C_{1}(d(y,z)+\gamma^{s(z,y^{\prime})})\leq C_{1}C_{4}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}),

as required. ∎

Lemma 8.3

If conditions (8.1) and  (8.2) are satisfied, then condition (H) holds.

Proof.

By (7.2) and (8.1), for all yY𝑦𝑌y\in Y, n0𝑛0n\geq 0,

|φ(Fnπy)φ(Fny)|C1d(Fnπy,Fny)C1C2γnd(πy,y)C1C2γn.𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦subscript𝐶1𝑑superscript𝐹𝑛𝜋𝑦superscript𝐹𝑛𝑦subscript𝐶1subscript𝐶2superscript𝛾𝑛𝑑𝜋𝑦𝑦subscript𝐶1subscript𝐶2superscript𝛾𝑛|\varphi(F^{n}\pi y)-\varphi(F^{n}y)|\leq C_{1}d(F^{n}\pi y,F^{n}y)\leq C_{1}C_{2}\gamma^{n}d(\pi y,y)\leq C_{1}C_{2}\gamma^{n}.

It follows that

|χ(y)|n=0|φ(Fnπy)φ(Fny)|C1C2(1γ)1.𝜒𝑦superscriptsubscript𝑛0𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦subscript𝐶1subscript𝐶2superscript1𝛾1\textstyle|\chi(y)|\leq\sum_{n=0}^{\infty}|\varphi(F^{n}\pi y)-\varphi(F^{n}y)|\leq C_{1}C_{2}(1-\gamma)^{-1}.

Hence |χ|C1C2(1γ)1subscript𝜒subscript𝐶1subscript𝐶2superscript1𝛾1|\chi|_{\infty}\leq C_{1}C_{2}(1-\gamma)^{-1} and condition (H)(a) is satisfied.

Next, let y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, and set N=[12s(y,y)]𝑁delimited-[]12𝑠𝑦superscript𝑦N=[\frac{1}{2}s(y,y^{\prime})], γ1=γ1/2subscript𝛾1superscript𝛾12\gamma_{1}=\gamma^{1/2}. Write

χ(y)χ(y)=A(πy,πy)A(y,y)+B(y)B(y),𝜒𝑦𝜒superscript𝑦𝐴𝜋𝑦𝜋superscript𝑦𝐴𝑦superscript𝑦𝐵𝑦𝐵superscript𝑦\chi(y)-\chi(y^{\prime})=A(\pi y,\pi y^{\prime})-A(y,y^{\prime})+B(y)-B(y^{\prime}),

where

A(p,q)=n=0N1(φ(Fnp)φ(Fnq)),B(p)=n=N(φ(Fnπp)φ(Fnp)).formulae-sequence𝐴𝑝𝑞superscriptsubscript𝑛0𝑁1𝜑superscript𝐹𝑛𝑝𝜑superscript𝐹𝑛𝑞𝐵𝑝superscriptsubscript𝑛𝑁𝜑superscript𝐹𝑛𝜋𝑝𝜑superscript𝐹𝑛𝑝A(p,q)=\sum_{n=0}^{N-1}(\varphi(F^{n}p)-\varphi(F^{n}q)),\qquad B(p)=\sum_{n=N}^{\infty}(\varphi(F^{n}\pi p)-\varphi(F^{n}p)).

By the calculation for |χ|subscript𝜒|\chi|_{\infty}, we obtain |B(p)|C1C2(1γ)1γN𝐵𝑝subscript𝐶1subscript𝐶2superscript1𝛾1superscript𝛾𝑁|B(p)|\leq C_{1}C_{2}(1-\gamma)^{-1}\gamma^{N} for all pY𝑝𝑌p\in Y. Also, γNγ1γ12s(y,y)=γ1γ1s(y,y)superscript𝛾𝑁superscript𝛾1superscript𝛾12𝑠𝑦superscript𝑦superscript𝛾1superscriptsubscript𝛾1𝑠𝑦superscript𝑦\gamma^{N}\leq\gamma^{-1}\gamma^{\frac{1}{2}s(y,y^{\prime})}=\gamma^{-1}\gamma_{1}^{s(y,y^{\prime})}, so B(p)=O(γ1s(y,y))𝐵𝑝𝑂superscriptsubscript𝛾1𝑠𝑦superscript𝑦B(p)=O(\gamma_{1}^{s(y,y^{\prime})}) for p=y,y𝑝𝑦superscript𝑦p=y,y^{\prime}.

For nN1𝑛𝑁1n\leq N-1 we have s(Fny,Fny)1𝑠superscript𝐹𝑛𝑦superscript𝐹𝑛superscript𝑦1s(F^{n}y,F^{n}y^{\prime})\geq 1, so by Propositions 7.3 and 8.2,

|φ(Fny)φ(Fny)|𝜑superscript𝐹𝑛𝑦𝜑superscript𝐹𝑛superscript𝑦\displaystyle|\varphi(F^{n}y)-\varphi(F^{n}y^{\prime})| C1C4(d(Fny,Fny)+γs(y,y)n)C(γnd(y,y)+γs(y,y)n),absentsubscript𝐶1subscript𝐶4𝑑superscript𝐹𝑛𝑦superscript𝐹𝑛superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝑛𝐶superscript𝛾𝑛𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝑛\displaystyle\leq C_{1}C_{4}(d(F^{n}y,F^{n}y^{\prime})+\gamma^{s(y,y^{\prime})-n})\leq C(\gamma^{n}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-n}),

where C=2C42C1C2𝐶2superscriptsubscript𝐶42subscript𝐶1subscript𝐶2C=2C_{4}^{2}C_{1}C_{2}. Hence

|A(y,y)|𝐴𝑦superscript𝑦\displaystyle|A(y,y^{\prime})| n=0N1|φ(Fny)φ(Fny)|Cn=0N1(γnd(y,y)+γs(y,y)n)absentsuperscriptsubscript𝑛0𝑁1𝜑superscript𝐹𝑛𝑦𝜑superscript𝐹𝑛superscript𝑦𝐶superscriptsubscript𝑛0𝑁1superscript𝛾𝑛𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝑛\displaystyle\leq\sum_{n=0}^{N-1}|\varphi(F^{n}y)-\varphi(F^{n}y^{\prime})|\leq C\sum_{n=0}^{N-1}(\gamma^{n}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-n})
C(1γ)1(d(y,y)+γs(y,y)N)C(1γ)1(d(y,y)+γ1s(y,y)).absent𝐶superscript1𝛾1𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦𝑁𝐶superscript1𝛾1𝑑𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦\displaystyle\leq C(1-\gamma)^{-1}(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-N})\leq C(1-\gamma)^{-1}(d(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}).

Similarly for A(πy,πy)𝐴𝜋𝑦𝜋superscript𝑦A(\pi y,\pi y^{\prime}). Hence |χ(y)χ(y)|d(y,y)+γ1s(y,y)much-less-than𝜒𝑦𝜒superscript𝑦𝑑𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦|\chi(y)-\chi(y^{\prime})|\ll d(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})}, so (H)(b) holds. ∎

8.2 Exponential contraction along stable leaves

In this subsection, we suppose that hCη(X)superscript𝐶𝜂𝑋h\in C^{\eta}(X) and that f𝑓f is exponentially contracting along stable leaves:

d(fny,fny)C2γnd(y,y)for all n0 and all y,yY with yWs(y).𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦subscript𝐶2superscript𝛾𝑛𝑑𝑦superscript𝑦for all n0 and all y,yY with yWs(y)d(f^{n}y,f^{n}y^{\prime})\leq C_{2}\gamma^{n}d(y,y^{\prime})\quad\text{for all $n\geq 0$ and all $y,y^{\prime}\in Y$ with $y^{\prime}\in W^{s}(y)$}. (8.3)

Note that this strengthens condition (7.2). Proposition 7.3 becomes

d(fny,fny)C2C4(γnd(y,y)+γs(y,y)ψn(y))for all n0y,yY.𝑑superscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦subscript𝐶2subscript𝐶4superscript𝛾𝑛𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦subscript𝜓𝑛𝑦for all n0y,yY.d(f^{n}y,f^{n}y^{\prime})\leq C_{2}C_{4}(\gamma^{n}d(y,y^{\prime})+\gamma^{s(y,y^{\prime})-\psi_{n}(y)})\quad\text{for all $n\geq 0$, $\,y,y^{\prime}\in Y$.} (8.4)
Lemma 8.4

If condition (8.3) is satisfied, then condition (H) holds.

Proof.

Let γ1=γηsubscript𝛾1superscript𝛾𝜂\gamma_{1}=\gamma^{\eta}, γ2=γ11/2subscript𝛾2superscriptsubscript𝛾112\gamma_{2}=\gamma_{1}^{1/2}. We verify condition (H) with γ2subscript𝛾2\gamma_{2} and d1(y,y)=d(y,y)ηsubscript𝑑1𝑦superscript𝑦𝑑superscript𝑦superscript𝑦𝜂d_{1}(y,y^{\prime})=d(y,y^{\prime})^{\eta}, using the equivalent definition for χ𝜒\chi,

χ(y)=n=0(h(fnπy)h(fny)).𝜒𝑦superscriptsubscript𝑛0superscript𝑓𝑛𝜋𝑦superscript𝑓𝑛𝑦\textstyle\chi(y)=\sum_{n=0}^{\infty}(h(f^{n}\pi y)-h(f^{n}y)).

By (8.3),

|χ(y)|n=0|h|ηd(fnπy,fny)ηC2|h|ηn=0γ1nd1(πy,y)C2|h|η(1γ1)1.𝜒𝑦superscriptsubscript𝑛0subscript𝜂𝑑superscriptsuperscript𝑓𝑛𝜋𝑦superscript𝑓𝑛𝑦𝜂subscript𝐶2subscript𝜂superscriptsubscript𝑛0superscriptsubscript𝛾1𝑛subscript𝑑1𝜋𝑦𝑦subscript𝐶2subscript𝜂superscript1subscript𝛾11\textstyle|\chi(y)|\leq\sum_{n=0}^{\infty}|h|_{\eta}d(f^{n}\pi y,f^{n}y)^{\eta}\leq C_{2}|h|_{\eta}\sum_{n=0}^{\infty}\gamma_{1}^{n}d_{1}(\pi y,y)\leq C_{2}|h|_{\eta}(1-\gamma_{1})^{-1}.

Hence |χ|C2|h|η(1γ1)1subscript𝜒subscript𝐶2subscript𝜂superscript1subscript𝛾11|\chi|_{\infty}\leq C_{2}|h|_{\eta}(1-\gamma_{1})^{-1} and condition (H)(a) is satisfied.

Next, let y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y and set N=[12s(y,y)]𝑁delimited-[]12𝑠𝑦superscript𝑦N=[\frac{1}{2}s(y,y^{\prime})]. Write χ(y)χ(y)=A(πy,πy)A(y,y)+B(y)B(y)𝜒𝑦𝜒superscript𝑦𝐴𝜋𝑦𝜋superscript𝑦𝐴𝑦superscript𝑦𝐵𝑦𝐵superscript𝑦\chi(y)-\chi(y^{\prime})=A(\pi y,\pi y^{\prime})-A(y,y^{\prime})+B(y)-B(y^{\prime}), where

A(p,q)=n=0N1(h(fnp)h(fnq)),B(p)=n=N(h(fnπp)h(fnp)).formulae-sequence𝐴𝑝𝑞superscriptsubscript𝑛0𝑁1superscript𝑓𝑛𝑝superscript𝑓𝑛𝑞𝐵𝑝superscriptsubscript𝑛𝑁superscript𝑓𝑛𝜋𝑝superscript𝑓𝑛𝑝A(p,q)=\sum_{n=0}^{N-1}(h(f^{n}p)-h(f^{n}q)),\qquad B(p)=\sum_{n=N}^{\infty}(h(f^{n}\pi p)-h(f^{n}p)).

By the calculation for |χ|subscript𝜒|\chi|_{\infty}, we obtain |B(p)|C2|h|η(1γ1)1γ1N𝐵𝑝subscript𝐶2subscript𝜂superscript1subscript𝛾11superscriptsubscript𝛾1𝑁|B(p)|\leq C_{2}|h|_{\eta}(1-\gamma_{1})^{-1}\gamma_{1}^{N} for all pY𝑝𝑌p\in Y. Also, γ1Nγ11γ112s(y,y)=γ11γ2s(y,y)superscriptsubscript𝛾1𝑁superscriptsubscript𝛾11superscriptsubscript𝛾112𝑠𝑦superscript𝑦superscriptsubscript𝛾11superscriptsubscript𝛾2𝑠𝑦superscript𝑦\gamma_{1}^{N}\leq\gamma_{1}^{-1}\gamma_{1}^{\frac{1}{2}s(y,y^{\prime})}=\gamma_{1}^{-1}\gamma_{2}^{s(y,y^{\prime})}, so B(p)=O(γ2s(y,y))𝐵𝑝𝑂superscriptsubscript𝛾2𝑠𝑦superscript𝑦B(p)=O(\gamma_{2}^{s(y,y^{\prime})}) for p=y,y𝑝𝑦superscript𝑦p=y,\,y^{\prime}.

Finally, by (8.4) using that ψnnsubscript𝜓𝑛𝑛\psi_{n}\leq n,

|A(y,y)|𝐴𝑦superscript𝑦\displaystyle|A(y,y^{\prime})| |h|ηn=0N1d(fny,fny)ηC2C4|h|ηn=0N1(γ1nd1(y,y)+γ1s(y,y)n)absentsubscript𝜂superscriptsubscript𝑛0𝑁1𝑑superscriptsuperscript𝑓𝑛𝑦superscript𝑓𝑛superscript𝑦𝜂subscript𝐶2subscript𝐶4subscript𝜂superscriptsubscript𝑛0𝑁1superscriptsubscript𝛾1𝑛subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑛\displaystyle\leq|h|_{\eta}\sum_{n=0}^{N-1}d(f^{n}y,f^{n}y^{\prime})^{\eta}\leq C_{2}C_{4}|h|_{\eta}\sum_{n=0}^{N-1}(\gamma_{1}^{n}d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})-n})
C2C4|h|η(1γ1)1(d1(y,y)+γ1s(y,y)N)absentsubscript𝐶2subscript𝐶4subscript𝜂superscript1subscript𝛾11subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾1𝑠𝑦superscript𝑦𝑁\displaystyle\leq C_{2}C_{4}|h|_{\eta}(1-\gamma_{1})^{-1}(d_{1}(y,y^{\prime})+\gamma_{1}^{s(y,y^{\prime})-N})
C2C4|h|η(1γ1)1(d1(y,y)+γ2s(y,y)).absentsubscript𝐶2subscript𝐶4subscript𝜂superscript1subscript𝛾11subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦\displaystyle\leq C_{2}C_{4}|h|_{\eta}(1-\gamma_{1})^{-1}(d_{1}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}).

Similarly for A(πy,πy)𝐴𝜋𝑦𝜋superscript𝑦A(\pi y,\pi y^{\prime}). Hence |χ(y)χ(y)|d1(y,y)+γ2s(y,y)much-less-than𝜒𝑦𝜒superscript𝑦subscript𝑑1𝑦superscript𝑦superscriptsubscript𝛾2𝑠𝑦superscript𝑦|\chi(y)-\chi(y^{\prime})|\ll d_{1}(y,y^{\prime})+\gamma_{2}^{s(y,y^{\prime})}, so (H)(b) holds. ∎

Remark 8.5

In cases where hh lies in Cη(X)superscript𝐶𝜂𝑋C^{\eta}(X) and the dynamics on X𝑋X is modelled by a Young tower with exponential tails (so μX(τ>n)=O(ect)subscript𝜇𝑋𝜏𝑛𝑂superscript𝑒𝑐𝑡\mu_{X}(\tau>n)=O(e^{-ct}) for some c>0𝑐0c>0), it is immediate that φLq(Y)𝜑superscript𝐿𝑞𝑌\varphi\in L^{q}(Y) for all q𝑞q and that condition (8.3) is satisfied. Assuming absence of approximate eigenfunctions, we obtain rapid mixing for such flows.

8.3 Flows with an invariant Hölder stable foliation

Let Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M be a Hölder nonuniformly hyperbolic flow as in Section 7.2. For simplicity, we suppose that (M,d)𝑀𝑑(M,d) is a Riemannian manifold and that Y𝑌Y is a smoothly embedded cross-section for the flow. We assume that the flow possesses a Ttsubscript𝑇𝑡T_{t}-invariant Hölder stable foliation 𝒲sssuperscript𝒲𝑠𝑠{\mathcal{W}}^{ss} in a neighbourhood of ΛΛ\Lambda. (A sufficient condition for this to hold is that ΛΛ\Lambda is a partially hyperbolic attracting set with a DTt𝐷subscript𝑇𝑡DT_{t}-invariant dominated splitting TΛM=EssEcusubscript𝑇Λ𝑀direct-sumsuperscript𝐸𝑠𝑠superscript𝐸𝑐𝑢T_{\Lambda}M=E^{ss}\oplus E^{cu}, see [3].) We also assume that diamYdiam𝑌\operatorname{diam}Y can be chosen arbitrarily small. In this subsection, we show how to use the stable foliation 𝒲sssuperscript𝒲𝑠𝑠{\mathcal{W}}^{ss} for the flow to show that χ𝜒\chi is Hölder, hence verifying the hypotheses in Section 6.1.

Remark 8.6

As discussed in [26, Section 4.2(iii)], this framework includes (not necessarily Markovian) intermittent solenoidal flows, and yields polynomial decay O(t(β1))𝑂superscript𝑡𝛽1O(t^{-(\beta-1)}) for any prescribed β>1𝛽1\beta>1. These results are optimal by [27] in the Markovian case and by [8] in general.

First, we show that if Ws(y)superscript𝑊𝑠𝑦W^{s}(y) and Wss(y)superscript𝑊𝑠𝑠𝑦W^{ss}(y) coincide for all yY𝑦𝑌y\in Y, then Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is already a skew product (so χ=0𝜒0\chi=0).

Proposition 8.7

Suppose that Ws(y)superscript𝑊𝑠𝑦W^{s}(y) and Wss(y)superscript𝑊𝑠𝑠𝑦W^{ss}(y) coincide for all yY𝑦𝑌y\in Y. Then φ𝜑\varphi is constant along stable leaves Ws(y)superscript𝑊𝑠𝑦W^{s}(y), yY𝑦𝑌y\in Y.

Proof.

For y0Ysubscript𝑦0𝑌y_{0}\in Y,

{Tφ(y)y:yWss(y0)}={Fy:yWs(y0)}=FWs(y0)Ws(Fy0)=Wss(Fy0).conditional-setsubscript𝑇𝜑𝑦𝑦𝑦superscript𝑊𝑠𝑠subscript𝑦0conditional-set𝐹𝑦𝑦superscript𝑊𝑠subscript𝑦0𝐹superscript𝑊𝑠subscript𝑦0superscript𝑊𝑠𝐹subscript𝑦0superscript𝑊𝑠𝑠𝐹subscript𝑦0\{T_{\varphi(y)}y:y\in W^{ss}(y_{0})\}=\{Fy:y\in W^{s}(y_{0})\}=FW^{s}(y_{0})\subset W^{s}(Fy_{0})=W^{ss}(Fy_{0}).

But setting t0=φ(y0)subscript𝑡0𝜑subscript𝑦0t_{0}=\varphi(y_{0}),

{Tt0y:yWss(y0)}=Tt0Wss(y0)Wss(Tt0y0)=Wss(Fy0).conditional-setsubscript𝑇subscript𝑡0𝑦𝑦superscript𝑊𝑠𝑠subscript𝑦0subscript𝑇subscript𝑡0superscript𝑊𝑠𝑠subscript𝑦0superscript𝑊𝑠𝑠subscript𝑇subscript𝑡0subscript𝑦0superscript𝑊𝑠𝑠𝐹subscript𝑦0\{T_{t_{0}}y:y\in W^{ss}(y_{0})\}=T_{t_{0}}W^{ss}(y_{0})\subset W^{ss}(T_{t_{0}}y_{0})=W^{ss}(Fy_{0}).

Hence φ|Wss(y0)φ(y0)evaluated-at𝜑superscript𝑊𝑠𝑠subscript𝑦0𝜑subscript𝑦0\varphi|_{W^{ss}(y_{0})}\equiv\varphi(y_{0}). ∎

Let Y~=Wu(y0)~𝑌superscript𝑊𝑢subscript𝑦0{\widetilde{Y}}=W^{u}(y_{0}) for some fixed y0Ysubscript𝑦0𝑌y_{0}\in Y and define the new cross-section to the flow Y=yY~Wss(y)superscript𝑌subscript𝑦~𝑌superscript𝑊𝑠𝑠𝑦Y^{*}=\bigcup_{y\in{\widetilde{Y}}}W^{ss}(y). Shrinking Y𝑌Y if necessary, there exists a unique continuous function r:Y:𝑟𝑌r:Y\to{\mathbb{R}} with |r|12infφ𝑟12infimum𝜑|r|\leq\frac{1}{2}\inf\varphi such that r|Y~0evaluated-at𝑟~𝑌0r|_{{\widetilde{Y}}}\equiv 0 and {Tr(y)(y):yY}Yconditional-setsubscript𝑇𝑟𝑦𝑦𝑦𝑌superscript𝑌\{T_{r(y)}(y):y\in Y\}\subset Y^{*}. Moreover, r𝑟r is Hölder since Y𝑌Y is smoothly embedded in M𝑀M and Ysuperscript𝑌Y^{*} is Hölder by the assumption on the regularity of the stable foliation 𝒲sssuperscript𝒲𝑠𝑠{\mathcal{W}}^{ss}. Define the new roof function

φ:Y+,φ(Tr(y)y)=φ(y)+r(Fy)r(y).:superscript𝜑formulae-sequencesuperscript𝑌superscriptsuperscript𝜑subscript𝑇𝑟𝑦𝑦𝜑𝑦𝑟𝐹𝑦𝑟𝑦\varphi^{*}:Y^{*}\to{\mathbb{R}}^{+},\qquad\varphi^{*}(T_{r(y)}y)=\varphi(y)+r(Fy)-r(y).

We observe that φsuperscript𝜑\varphi^{*} is the return time for the flow Ttsubscript𝑇𝑡T_{t} to the cross-section Ysuperscript𝑌Y^{*}.

Lemma 8.8

Under the above assumption on 𝒲sssuperscript𝒲𝑠𝑠{\mathcal{W}}^{ss}, condition (H) holds.

Proof.

We show that χ=r𝜒𝑟\chi=-r. The result follows since r𝑟r is Hölder.

Let n0𝑛0n\geq 0, yY𝑦𝑌y\in Y. By Proposition 8.7 applied to φ:Y+:superscript𝜑superscript𝑌superscript\varphi^{*}:Y^{*}\to{\mathbb{R}}^{+}, we have that φ(Tr(Fnπy)Fnπy)=φ(Tr(Fny)Fny)superscript𝜑subscript𝑇𝑟superscript𝐹𝑛𝜋𝑦superscript𝐹𝑛𝜋𝑦superscript𝜑subscript𝑇𝑟superscript𝐹𝑛𝑦superscript𝐹𝑛𝑦\varphi^{*}(T_{r(F^{n}\pi y)}F^{n}\pi y)=\varphi^{*}(T_{r(F^{n}y)}F^{n}y). Hence by definition of φsuperscript𝜑\varphi^{*},

φ(Fnπy)φ(Fny)=r(Fnπy)r(Fny)+r(Fn+1y)r(Fn+1πy).𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦𝑟superscript𝐹𝑛𝜋𝑦𝑟superscript𝐹𝑛𝑦𝑟superscript𝐹𝑛1𝑦𝑟superscript𝐹𝑛1𝜋𝑦\varphi(F^{n}\pi y)-\varphi(F^{n}y)=r(F^{n}\pi y)-r(F^{n}y)+r(F^{n+1}y)-r(F^{n+1}\pi y).

Let η𝜂\eta be the Hölder exponent of r𝑟r. By (7.2), |φ(Fnπy)φ(Fny)|2C2|r|η(γη)n𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦2subscript𝐶2subscript𝑟𝜂superscriptsuperscript𝛾𝜂𝑛|\varphi(F^{n}\pi y)-\varphi(F^{n}y)|\leq 2C_{2}|r|_{\eta}(\gamma^{\eta})^{n} so the series χ(y)=n=0(φ(Fnπy)φ(Fny))𝜒𝑦superscriptsubscript𝑛0𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦\chi(y)=\sum_{n=0}^{\infty}(\varphi(F^{n}\pi y)-\varphi(F^{n}y)) converges absolutely. Moreover,

χ(y)𝜒𝑦\displaystyle\chi(y) =limNn=0N1(φ(Fnπy)φ(Fny))absentsubscript𝑁superscriptsubscript𝑛0𝑁1𝜑superscript𝐹𝑛𝜋𝑦𝜑superscript𝐹𝑛𝑦\displaystyle=\lim_{N\to\infty}\sum_{n=0}^{N-1}(\varphi(F^{n}\pi y)-\varphi(F^{n}y))
=limN(r(πy)r(y)+r(FNy)r(FNπy))=r(πy)r(y).absentsubscript𝑁𝑟𝜋𝑦𝑟𝑦𝑟superscript𝐹𝑁𝑦𝑟superscript𝐹𝑁𝜋𝑦𝑟𝜋𝑦𝑟𝑦\displaystyle=\lim_{N\to\infty}\big{(}r(\pi y)-r(y)+r(F^{N}y)-r(F^{N}\pi y)\big{)}=r(\pi y)-r(y).

Finally, r(πy)=0𝑟𝜋𝑦0r(\pi y)=0 since r|Y~0evaluated-at𝑟~𝑌0r|_{{\widetilde{Y}}}\equiv 0. ∎

8.4 Temporal distance function

Dolgopyat [18, Appendix] showed that for Axiom A flows a sufficient condition for absence of approximate eigenfunctions is that the range of the temporal distance function has positive lower box dimension. This was extended to nonuniformly hyperbolic flows in [25, 26]. Here we recall the main definitions and result.

We assume that condition (H) holds, so that the suspension flow YφYφsuperscript𝑌𝜑superscript𝑌𝜑Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow (and hence conjugate to a skew product flow). We also assume the dynamically Hölder setup from Section 7.3. In particular, the Poincaré map f:XX:𝑓𝑋𝑋f:X\to X is nonuniformly hyperbolic as in Section 7.1 and Y𝑌Y has a local product structure. Also we assume that the roof function φ𝜑\varphi has bounded Hölder constants along unstable leaves, so condition (8.2) is satisfied.

Let y1,y4Ysubscript𝑦1subscript𝑦4𝑌y_{1},y_{4}\in Y and set y2=Ws(y1)Wu(y4)subscript𝑦2superscript𝑊𝑠subscript𝑦1superscript𝑊𝑢subscript𝑦4y_{2}=W^{s}(y_{1})\cap W^{u}(y_{4}), y3=Wu(y1)Ws(y4)subscript𝑦3superscript𝑊𝑢subscript𝑦1superscript𝑊𝑠subscript𝑦4y_{3}=W^{u}(y_{1})\cap W^{s}(y_{4}). Define the temporal distance function D:Y×Y:𝐷𝑌𝑌D:Y\times Y\to{\mathbb{R}},

D(y1,y4)=n=(φ(Fny1)φ(Fny2)φ(Fny3)+φ(Fny4)).𝐷subscript𝑦1subscript𝑦4superscriptsubscript𝑛𝜑superscript𝐹𝑛subscript𝑦1𝜑superscript𝐹𝑛subscript𝑦2𝜑superscript𝐹𝑛subscript𝑦3𝜑superscript𝐹𝑛subscript𝑦4D(y_{1},y_{4})=\sum_{n=-\infty}^{\infty}\Big{(}\varphi(F^{n}y_{1})-\varphi(F^{n}y_{2})-\varphi(F^{n}y_{3})+\varphi(F^{n}y_{4})\Big{)}.

It follows from the construction in [26, Section 5.3] (which uses (7.4) and (8.2)) that inverse branches Fnyisuperscript𝐹𝑛subscript𝑦𝑖F^{n}y_{i} for n1𝑛1n\leq-1 can be chosen so that D𝐷D is well-defined.

Lemma 8.9 ( [26, Theorem 5.6])

Let Z0=n=0FnZsubscript𝑍0superscriptsubscript𝑛0superscript𝐹𝑛𝑍Z_{0}=\bigcap_{n=0}^{\infty}F^{-n}Z where Z𝑍Z is a union of finitely many elements of the partition {Yj}subscript𝑌𝑗\{Y_{j}\}. Let Z¯0subscript¯𝑍0{\mkern 1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu}_{0} denote the corresponding finite subsystem of Y¯¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu}. If the lower box dimension of D(Z0×Z0)𝐷subscript𝑍0subscript𝑍0D(Z_{0}\times Z_{0}) is positive, then there do not exist approximate eigenfunctions on Z¯0subscript¯𝑍0{\mkern 1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu}_{0}. ∎

Remark 8.10

For Axiom A attractors, Z0subscript𝑍0Z_{0} can be taken to be connected and D𝐷D is continuous, so absence of approximate eigenfunctions is ensured whenever D𝐷D is not identically zero. For nonuniformly hyperbolic flows, where the partition {Yj}subscript𝑌𝑗\{Y_{j}\} is countably infinite, Z0subscript𝑍0Z_{0} is a Cantor set of positive Hausdorff dimension [25, Example 5.7]. In general it is not clear how to use this property since D𝐷D is generally at best Hölder. However for flows with a contact structure, a formula for D𝐷D in [21, Lemma 3.2] can be exploited and the lower box dimension of D(Z0×Z0)𝐷subscript𝑍0subscript𝑍0D(Z_{0}\times Z_{0}) is indeed positive, see [25, Example 5.7]. The arguments in [25, Example 5.7] apply to general Gibbs-Markov flows with a contact structure. A special case of this is the Lorentz gas examples considered in Section 9.

9 Billiard flows associated to infinite horizon Lorentz gases

In this section we show that billiard flows associated to planar infinite horizon Lorentz gases satisfy the assumptions of Section 8.1. In particular, we prove decay of correlations with decay rate O(t1)𝑂superscript𝑡1O(t^{-1}).

Background material on infinite horizon Lorentz gases is recalled in Subsection 9.1 and the decay rate O(t1)𝑂superscript𝑡1O(t^{-1}) is proved in Subsection 9.2. In Subsection 9.3, we show that the same decay rate holds for semidispersing Lorentz flows and stadia. In Subsection 9.4, we show that the decay rate is optimal for the examples considered in this section.

9.1 Background on the infinite horizon Lorentz gas

We begin by recalling some background on billiard flows; for further details we refer to the monograph [13].

Let 𝕋2superscript𝕋2{\mathbb{T}}^{2} denote the two dimensional flat torus, and let us fix finitely many disjoint convex scatterers Sk𝕋2subscript𝑆𝑘superscript𝕋2S_{k}\subset{\mathbb{T}}^{2} with C3superscript𝐶3C^{3} boundaries of nonvanishing curvature. The complement Q=𝕋2Sk𝑄superscript𝕋2subscript𝑆𝑘Q={\mathbb{T}}^{2}\setminus\bigcup S_{k} is the billiard domain, and the billiard dynamics are that of a point particle that performs uniform motion with unit speed inside Q𝑄Q, and specular reflections — angle of reflection equals angle of incidence — off the scatterers, that is, at the boundary Q𝑄\partial Q. The resulting billiard flow is Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M, where the phase space M=Q×𝕊1𝑀𝑄superscript𝕊1M=Q\times{\mathbb{S}}^{1} is a Riemannian manifold, and Ttsubscript𝑇𝑡T_{t} preserves the (normalized) Lebesgue measure μMsubscript𝜇𝑀\mu_{M} (often called Liouville measure in the literature).

There is a natural Poincaré section X=Q×[π/2,π/2]M𝑋𝑄𝜋2𝜋2𝑀X=\partial Q\times[-\pi/2,\pi/2]\subset M corresponding to collisions (with outgoing velocities), which gives rise to the billiard map denoted by f:XX:𝑓𝑋𝑋f:X\to X, with absolute continuous invariant probability measure μXsubscript𝜇𝑋\mu_{X}. The time until the next collision, the free flight function h:X+:𝑋superscripth:X\to{\mathbb{R}}^{+}, is defined to be h(x)=inf{t>0:TtxX}𝑥infimumconditional-set𝑡0subscript𝑇𝑡𝑥𝑋h(x)=\inf\{t>0:T_{t}x\in X\}. The Lorentz gas has finite horizon if hL(X)superscript𝐿𝑋h\in L^{\infty}(X) and infinite horizon if hh is unbounded.

In the finite horizon case, [4] recently proved exponential decay of correlations. In this section, we prove

Theorem 9.1

Let η(0,1]𝜂01\eta\in(0,1]. In the infinite horizon case, there exists m1𝑚1m\geq 1 such that ρv,w(t)=O(t1)subscript𝜌𝑣𝑤𝑡𝑂superscript𝑡1\rho_{v,w}(t)=O(t^{-1}) for all vCη(M)C0,η(M)𝑣superscript𝐶𝜂𝑀superscript𝐶0𝜂𝑀v\in C^{\eta}(M)\cap C^{0,\eta}(M) and wCη,m(M)𝑤superscript𝐶𝜂𝑚𝑀w\in C^{\eta,m}(M) (and more generally for the class of observables defined in Corollary 9.6 below).

Let us fix some terminology and notations. The billiard map f:XX:𝑓𝑋𝑋f:X\to X is discontinuous, with singularity set 𝒮𝒮{\mathcal{S}} corresponding to the preimages of grazing collisions. Here, 𝒮𝒮{\mathcal{S}} is the closure of a countable union of smooth curves, X𝒮𝑋𝒮X\setminus{\mathcal{S}} consists of countably many connected components Xmsubscript𝑋𝑚X_{m}, m1𝑚1m\geq 1, and f|Xmevaluated-at𝑓subscript𝑋𝑚f|_{X_{m}} is C2superscript𝐶2C^{2}. If x,xXm𝑥superscript𝑥subscript𝑋𝑚x,x^{\prime}\in X_{m} for some m1𝑚1m\geq 1, then, in particular, x,x𝑥superscript𝑥x,x^{\prime} and fx,fx𝑓𝑥𝑓superscript𝑥fx,fx^{\prime} lie on the same scatterer (even when the configuration is unfolded to the plane). Throughout our exposition, d(x,x)𝑑𝑥superscript𝑥d(x,x^{\prime}) denotes the Euclidean distance of the two points, i.e. the distance that is generated by the Riemannian metric on X𝑋X (or M𝑀M).

It follows from geometric considerations in the infinite horizon case that μX(h>t)=O(t2)subscript𝜇𝑋𝑡𝑂superscript𝑡2\mu_{X}(h>t)=O(t^{-2}). Moreover, as the trajectories are straight lines, we have

|h(x)h(x)|d(x,x)+d(fx,fx)for all x,xXmm1; and𝑥superscript𝑥𝑑𝑥superscript𝑥𝑑𝑓𝑥𝑓superscript𝑥for all x,xXmm1; and\displaystyle|h(x)-h(x^{\prime})|\leq d(x,x^{\prime})+d(fx,fx^{\prime})\quad\text{for all $x,x^{\prime}\in X_{m}$, $m\geq 1$; and} (9.1)
d(Ttx,Ttx)|tt|for all xX and t,t[0,h(x)).𝑑subscript𝑇𝑡𝑥subscript𝑇superscript𝑡𝑥𝑡superscript𝑡for all xX and t,t[0,h(x))\displaystyle d(T_{t}x,T_{t^{\prime}}x)\leq|t-t^{\prime}|\quad\text{for all $x\in X$ and $t,t^{\prime}\in[0,h(x))$}. (9.2)

The billiard maps considered here (both finite and infinite horizon) have uniform contraction and expansion even for f𝑓f. There exist stable and unstable manifolds of positive length for almost every xX𝑥𝑋x\in X, which we denote by Ws(x)superscript𝑊𝑠𝑥W^{s}(x) and Wu(x)superscript𝑊𝑢𝑥W^{u}(x) respectively, and there exist constants C21subscript𝐶21C_{2}\geq 1, γ(0,1)𝛾01\gamma\in(0,1) such that for all x,xX𝑥superscript𝑥𝑋x,x^{\prime}\in X, n0𝑛0n\geq 0,

d(fnx,fnx)C2γnd(x,x)for xWs(x).𝑑superscript𝑓𝑛𝑥superscript𝑓𝑛superscript𝑥subscript𝐶2superscript𝛾𝑛𝑑𝑥superscript𝑥for xWs(x).\displaystyle d(f^{n}x,f^{n}x^{\prime})\leq C_{2}\gamma^{n}d(x,x^{\prime})\quad\text{for $x^{\prime}\in W^{s}(x)$.} (9.3)
d(x,x)C2γnd(fnx,fnx)for fnxWu(fnx).𝑑𝑥superscript𝑥subscript𝐶2superscript𝛾𝑛𝑑superscript𝑓𝑛𝑥superscript𝑓𝑛superscript𝑥for fnxWu(fnx).\displaystyle d(x,x^{\prime})\leq C_{2}\gamma^{n}d(f^{n}x,f^{n}x^{\prime})\quad\text{for $f^{n}x^{\prime}\in W^{u}(f^{n}x)$.} (9.4)

This follows from the uniform hyperbolicity properties of f𝑓f, see in particular [13, Formula (4.19)].

Furthermore, there is a constant C51subscript𝐶51C_{5}\geq 1 such that for x,xX𝑥superscript𝑥𝑋x,x^{\prime}\in X,

d(Ttx,Ttx)𝑑subscript𝑇𝑡𝑥subscript𝑇𝑡superscript𝑥\displaystyle d(T_{t}x,T_{t}x^{\prime}) C5d(x,x)for xWs(x)t[0,h(x)][0,h(x)].absentsubscript𝐶5𝑑𝑥superscript𝑥for xWs(x)t[0,h(x)][0,h(x)].\displaystyle\leq C_{5}d(x,x^{\prime})\quad\text{for $x^{\prime}\in W^{s}(x)$, $t\in[0,h(x)]\cap[0,h(x^{\prime})]$.} (9.5)
d(Ttx,Ttx)𝑑subscript𝑇𝑡𝑥subscript𝑇𝑡superscript𝑥\displaystyle d(T_{-t}x,T_{-t}x^{\prime}) C5d(x,x)for xWu(x)t[0,h(f1x)][0,h(f1x)].absentsubscript𝐶5𝑑𝑥superscript𝑥for xWu(x)t[0,h(f1x)][0,h(f1x)].\displaystyle\leq C_{5}d(x,x^{\prime})\quad\text{for $x^{\prime}\in W^{u}(x)$, $t\in[0,h(f^{-1}x)]\cap[0,h(f^{-1}x^{\prime})]$.} (9.6)

To verify (9.5), note that d(x,x)𝑑𝑥superscript𝑥d(x,x^{\prime}) consists of a position and a velocity component. In course of the free flight, the velocities do not change, while for xWs(x)superscript𝑥superscript𝑊𝑠𝑥x^{\prime}\in W^{s}(x), the position component can only shrink as stable manifolds correspond to converging wavefronts. A similar argument applies to (9.6).

Remark 9.2

(a) In the remainder of the section – and in particular in the proof of Proposition 9.5 below – we apply (9.1) repeatedly, but always in the case when either xWs(x)superscript𝑥superscript𝑊𝑠𝑥x^{\prime}\in W^{s}(x), or fxWu(fx)𝑓superscript𝑥superscript𝑊𝑢𝑓𝑥fx^{\prime}\in W^{u}(fx). As all iterates fn,n0superscript𝑓𝑛𝑛0f^{n},n\geq 0 are smooth on local stable manifolds (while all iterates fn,n0superscript𝑓𝑛𝑛0f^{-n},n\geq 0 are smooth on local unstable manifolds), both of these conditions imply x,xXm𝑥superscript𝑥subscript𝑋𝑚x,x^{\prime}\in X_{m} for some m1𝑚1m\geq 1.
(b) For larger values of t𝑡t than those in (9.5), we note that d(Ttx,Ttx)𝑑subscript𝑇𝑡𝑥subscript𝑇𝑡superscript𝑥d(T_{t}x,T_{t}x^{\prime}) may grow large temporarily: it can happen that one of the trajectories has already collided with some scatterer, while the other has not, hence even though the two points are close in position, the velocities differ substantially. Similar comments apply to (9.6). This phenomenon is the main reason why we require the notion of dynamically Hölder flows Ttsubscript𝑇𝑡T_{t} in Definition 7.6.

In [30], Young constructs a subset YX𝑌𝑋Y\subset X and an induced map F=fτ:YY:𝐹superscript𝑓𝜏𝑌𝑌F=f^{\tau}:Y\to Y that possesses the properties discussed in Section 7.1 including (7.4). The tails of the return time τ:Y+:𝜏𝑌superscript\tau:Y\to{\mathbb{Z}}^{+} are exponential, i.e. μ(τ>n)=O(ecn)𝜇𝜏𝑛𝑂superscript𝑒𝑐𝑛\mu(\tau>n)=O(e^{-cn}) for some c>0𝑐0c>0. Moreover, the construction can be carried out so that diamYdiam𝑌\operatorname{diam}Y is as small as desired. This is proved in [30] for the finite horizon, and in [11] for the infinite horizon case. We mention that (7.2) and (7.3) follow from (9.3) and (9.4), respectively, while (7.1) holds as the stable and the unstable manifolds are uniformly transversal, see [13, Formulas (4.13) and (4.21)].

Proposition 9.3

For all y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j}, j1𝑗1j\geq 1, and all 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1,

|h(fy)h(fy)|2C22C4γ1(γd(y,y)+γτ(y)γs(y,y)).superscript𝑓𝑦superscript𝑓superscript𝑦2superscriptsubscript𝐶22subscript𝐶4superscript𝛾1superscript𝛾𝑑𝑦superscript𝑦superscript𝛾𝜏𝑦superscript𝛾𝑠𝑦superscript𝑦|h(f^{\ell}y)-h(f^{\ell}y^{\prime})|\leq 2C_{2}^{2}C_{4}\gamma^{-1}(\gamma^{\ell}d(y,y^{\prime})+\gamma^{\tau(y)-\ell}\gamma^{s(y,y^{\prime})}).
Proof.

Let z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}). By (7.4), FzWu(Fy)𝐹𝑧superscript𝑊𝑢𝐹superscript𝑦Fz\in W^{u}(Fy^{\prime}). By (9.3) and (9.4), for 0τ(y)0𝜏𝑦0\leq\ell\leq\tau(y),

d(fy,fy)d(fy,fz)+d(fz,fy)C2(γd(y,z)+γτ(y)d(Fz,Fy)).𝑑superscript𝑓𝑦superscript𝑓superscript𝑦𝑑superscript𝑓𝑦superscript𝑓𝑧𝑑superscript𝑓𝑧superscript𝑓superscript𝑦subscript𝐶2superscript𝛾𝑑𝑦𝑧superscript𝛾𝜏𝑦𝑑𝐹𝑧𝐹superscript𝑦d(f^{\ell}y,f^{\ell}y^{\prime})\leq d(f^{\ell}y,f^{\ell}z)+d(f^{\ell}z,f^{\ell}y^{\prime})\leq C_{2}(\gamma^{\ell}d(y,z)+\gamma^{\tau(y)-\ell}d(Fz,Fy^{\prime})).

Using also (7.1) and (7.3),

d(fy,fy)C2(γC4d(y,y)+γτ(y)C2γs(y,y)1).𝑑superscript𝑓𝑦superscript𝑓superscript𝑦subscript𝐶2superscript𝛾subscript𝐶4𝑑𝑦superscript𝑦superscript𝛾𝜏𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦1d(f^{\ell}y,f^{\ell}y^{\prime})\leq C_{2}(\gamma^{\ell}C_{4}d(y,y^{\prime})+\gamma^{\tau(y)-\ell}C_{2}\gamma^{s(y,y^{\prime})-1}).

Hence by (9.1), for τ(y)1𝜏𝑦1\ell\leq\tau(y)-1,

|h(fy)h(fy)|d(fy,fy)+d(f+1y,f+1y)γd(y,y)+γτ(y)γs(y,y),superscript𝑓𝑦superscript𝑓superscript𝑦𝑑superscript𝑓𝑦superscript𝑓superscript𝑦𝑑superscript𝑓1𝑦superscript𝑓1superscript𝑦much-less-thansuperscript𝛾𝑑𝑦superscript𝑦superscript𝛾𝜏𝑦superscript𝛾𝑠𝑦superscript𝑦|h(f^{\ell}y)-h(f^{\ell}y^{\prime})|\leq d(f^{\ell}y,f^{\ell}y^{\prime})+d(f^{\ell+1}y,f^{\ell+1}y^{\prime})\ll\gamma^{\ell}d(y,y^{\prime})+\gamma^{\tau(y)-\ell}\gamma^{s(y,y^{\prime})},

as required. ∎

Define the induced roof function φ==0τ1hf𝜑superscriptsubscript0𝜏1superscript𝑓\varphi=\sum_{\ell=0}^{\tau-1}h\circ f^{\ell}. Using (7.3), it is immediate from Proposition 9.3 that φ𝜑\varphi has bounded Hölder constants in the sense of Section 8.1:

Corollary 9.4

Conditions (8.1) and (8.2) hold.

Proof.

If yWs(y)superscript𝑦superscript𝑊𝑠𝑦y^{\prime}\in W^{s}(y), then s(y,y)=𝑠𝑦superscript𝑦s(y,y^{\prime})=\infty so |φ(y)φ(y)|d(y,y)much-less-than𝜑𝑦𝜑superscript𝑦𝑑𝑦superscript𝑦|\varphi(y)-\varphi(y^{\prime})|\ll d(y,y^{\prime}) by Proposition 9.3. If yWu(y)superscript𝑦superscript𝑊𝑢𝑦y^{\prime}\in W^{u}(y), then d(y,y)C2γs(y,y)𝑑𝑦superscript𝑦subscript𝐶2superscript𝛾𝑠𝑦superscript𝑦d(y,y^{\prime})\leq C_{2}\gamma^{s(y,y^{\prime})} by (7.3), so |φ(y)φ(y)|γs(y,y)much-less-than𝜑𝑦𝜑superscript𝑦superscript𝛾𝑠𝑦superscript𝑦|\varphi(y)-\varphi(y^{\prime})|\ll\gamma^{s(y,y^{\prime})} by Proposition 9.3.  ∎

Proposition 9.5

For diamYdiam𝑌\operatorname{diam}Y sufficiently small, there exist an integer n01subscript𝑛01n_{0}\geq 1 and a constant C>0𝐶0C>0 such that for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, s(y,y)n0𝑠𝑦superscript𝑦subscript𝑛0s(y,y^{\prime})\geq n_{0}, and all u[0,φ(y)][0,φ(y)]𝑢0𝜑𝑦0𝜑superscript𝑦u\in[0,\varphi(y)]\cap[0,\varphi(y^{\prime})], there exist t,t𝑡superscript𝑡t,t^{\prime}\in{\mathbb{R}} such that

|tu|𝑡𝑢\displaystyle|t-u| Cd(y,y),absent𝐶𝑑𝑦superscript𝑦\displaystyle\leq Cd(y,y^{\prime}), d(Tuy,Ttz)𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧\displaystyle\qquad d(T_{u}y,T_{t}z) Cd(y,y),absent𝐶𝑑𝑦superscript𝑦\displaystyle\leq Cd(y,y^{\prime}),
|tu|superscript𝑡𝑢\displaystyle|t^{\prime}-u| Cγs(y,y),absent𝐶superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq C\gamma^{s(y,y^{\prime})}, d(Tuy,Ttz)𝑑subscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑡𝑧\displaystyle\qquad d(T_{u}y^{\prime},T_{t^{\prime}}z) Cγs(y,y),absent𝐶superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq C\gamma^{s(y,y^{\prime})},

where z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}).

Proof.

Define h(y)=j=01h(fjy)subscript𝑦superscriptsubscript𝑗01superscript𝑓𝑗𝑦h_{\ell}(y)=\sum_{j=0}^{\ell-1}h(f^{j}y) for yY𝑦𝑌y\in Y, 0τ(y)0𝜏𝑦0\leq\ell\leq\tau(y). By Proposition 9.3, there is a constant C>0𝐶0C>0 such that

|h(y)h(y)|j=0τ(y)1|h(fjy)h(fjy)|C(d(y,y)+γs(y,y)),subscript𝑦subscriptsuperscript𝑦superscriptsubscript𝑗0𝜏𝑦1superscript𝑓𝑗𝑦superscript𝑓𝑗superscript𝑦𝐶𝑑𝑦superscript𝑦superscript𝛾𝑠𝑦superscript𝑦|h_{\ell}(y)-h_{\ell}(y^{\prime})|\leq\sum_{j=0}^{\tau(y)-1}|h(f^{j}y)-h(f^{j}y^{\prime})|\leq C(d(y,y^{\prime})+\gamma^{s(y,y^{\prime})}), (9.7)

for all y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j}, j1𝑗1j\geq 1 (which is equivalent to s(y,y)1𝑠𝑦superscript𝑦1s(y,y^{\prime})\geq 1) and all 0τ(y)0𝜏𝑦0\leq\ell\leq\tau(y).

Now consider y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y with s(y,y)n0𝑠𝑦superscript𝑦subscript𝑛0s(y,y^{\prime})\geq n_{0}, and u[0,φ(y)][0,φ(y)]𝑢0𝜑𝑦0𝜑superscript𝑦u\in[0,\varphi(y)]\cap[0,\varphi(y^{\prime})]. Let z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}).

Choosing t𝑡t. By (7.1), d(y,z)C4d(y,y)𝑑𝑦𝑧subscript𝐶4𝑑𝑦superscript𝑦d(y,z)\leq C_{4}d(y,y^{\prime}). Also, s(y,z)=𝑠𝑦𝑧s(y,z)=\infty. We can shrink Y𝑌Y if necessary so that CdiamYinfh𝐶diam𝑌infimumC\operatorname{diam}Y\leq\inf h.

Write Tuy=Trfysubscript𝑇𝑢𝑦subscript𝑇𝑟superscript𝑓𝑦T_{u}y=T_{r}f^{\ell}y where 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1 and r[0,h(fy))𝑟0superscript𝑓𝑦r\in[0,h(f^{\ell}y)). (When u=φ(y)𝑢𝜑𝑦u=\varphi(y), we take =τ(y)1𝜏𝑦1\ell=\tau(y)-1, r=h(fy)𝑟superscript𝑓𝑦r=h(f^{\ell}y).) Similarly, write Tuz=Trfzsubscript𝑇𝑢𝑧subscript𝑇superscript𝑟superscript𝑓superscript𝑧T_{u}z=T_{r^{\prime}}f^{\ell^{\prime}}z. Note that u=h(y)+r=h(z)+r𝑢subscript𝑦𝑟subscriptsuperscript𝑧superscript𝑟u=h_{\ell}(y)+r=h_{\ell^{\prime}}(z)+r^{\prime}.

First we show that ||1superscript1|\ell-\ell^{\prime}|\leq 1. By (9.7),

(1)infhsuperscript1infimum\displaystyle(\ell-\ell^{\prime}-1)\inf h h(z)h+1(z)h(y)h+1(z)+h(z)h(y)absentsubscript𝑧subscriptsuperscript1𝑧subscript𝑦subscriptsuperscript1𝑧subscript𝑧subscript𝑦\displaystyle\leq h_{\ell}(z)-h_{\ell^{\prime}+1}(z)\leq h_{\ell}(y)-h_{\ell^{\prime}+1}(z)+h_{\ell}(z)-h_{\ell}(y)
h(y)h(z)h(fz)+CdiamYabsentsubscript𝑦subscriptsuperscript𝑧superscript𝑓superscript𝑧𝐶diam𝑌\displaystyle\leq h_{\ell}(y)-h_{\ell^{\prime}}(z)-h(f^{\ell^{\prime}}z)+C\operatorname{diam}Y
=rrh(fz)+CdiamYCdiamYinfh.absentsuperscript𝑟𝑟superscript𝑓superscript𝑧𝐶diam𝑌𝐶diam𝑌infimum\displaystyle=r^{\prime}-r-h(f^{\ell^{\prime}}z)+C\operatorname{diam}Y\leq C\operatorname{diam}Y\leq\inf h.

Hence +1superscript1\ell\leq\ell^{\prime}+1. Similarly, (1)infhh(y)h+1(y)infhsuperscript1infimumsubscriptsuperscript𝑦subscript1𝑦infimum(\ell^{\prime}-\ell-1)\inf h\leq h_{\ell^{\prime}}(y)-h_{\ell+1}(y)\leq\inf h, so ||1superscript1|\ell-\ell^{\prime}|\leq 1.

If =superscript\ell=\ell^{\prime}, then we take t=u𝑡𝑢t=u. By (9.7),

|rr|=|h(y)h(z)|Cd(y,z)CC4d(y,y).𝑟superscript𝑟subscript𝑦subscript𝑧𝐶𝑑𝑦𝑧𝐶subscript𝐶4𝑑𝑦superscript𝑦|r-r^{\prime}|=|h_{\ell}(y)-h_{\ell}(z)|\leq Cd(y,z)\leq CC_{4}d(y,y^{\prime}).

By (9.3), d(fy,fz)C2d(y,z)C2C4d(y,y)𝑑superscript𝑓𝑦superscript𝑓𝑧subscript𝐶2𝑑𝑦𝑧subscript𝐶2subscript𝐶4𝑑𝑦superscript𝑦d(f^{\ell}y,f^{\ell}z)\leq C_{2}d(y,z)\leq C_{2}C_{4}d(y,y^{\prime}). Without loss, rr𝑟superscript𝑟r\leq r^{\prime}, so by (9.2) and (9.5)

d(Tuy,Ttz)=d(Trfy,Trfz)𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧𝑑subscript𝑇𝑟superscript𝑓𝑦subscript𝑇superscript𝑟superscript𝑓𝑧\displaystyle d(T_{u}y,T_{t}z)=d(T_{r}f^{\ell}y,T_{r^{\prime}}f^{\ell}z) d(Trfy,Trfz)+d(Trfz,Trfz)absent𝑑subscript𝑇𝑟superscript𝑓𝑦subscript𝑇𝑟superscript𝑓𝑧𝑑subscript𝑇𝑟superscript𝑓𝑧subscript𝑇superscript𝑟superscript𝑓𝑧\displaystyle\leq d(T_{r}f^{\ell}y,T_{r}f^{\ell}z)+d(T_{r}f^{\ell}z,T_{r^{\prime}}f^{\ell}z)
C5d(fy,fz)+|rr|d(y,y).absentsubscript𝐶5𝑑superscript𝑓𝑦superscript𝑓𝑧𝑟superscript𝑟much-less-than𝑑𝑦superscript𝑦\displaystyle\leq C_{5}d(f^{\ell}y,f^{\ell}z)+|r-r^{\prime}|\ll d(y,y^{\prime}).

If =1superscript1\ell^{\prime}=\ell-1, then we take t=u+r+s𝑡𝑢𝑟𝑠t=u+r+s where s=h(f1z)r0𝑠superscript𝑓1𝑧superscript𝑟0s=h(f^{\ell-1}z)-r^{\prime}\geq 0. Then Tuy=Trfysubscript𝑇𝑢𝑦subscript𝑇𝑟superscript𝑓𝑦T_{u}y=T_{r}f^{\ell}y and Ttz=Tr+sTrf1z=Tr+h(f1z)f1z=Trfzsubscript𝑇𝑡𝑧subscript𝑇𝑟𝑠subscript𝑇superscript𝑟superscript𝑓1𝑧subscript𝑇𝑟superscript𝑓1𝑧superscript𝑓1𝑧subscript𝑇𝑟superscript𝑓𝑧T_{t}z=T_{r+s}T_{r^{\prime}}f^{\ell-1}z=T_{r+h(f^{\ell-1}z)}f^{\ell-1}z=T_{r}f^{\ell}z.

Note that u=h(y)+r=h(z)s𝑢subscript𝑦𝑟subscript𝑧𝑠u=h_{\ell}(y)+r=h_{\ell}(z)-s, hence r+s=h(z)h(y)Cd(y,z)𝑟𝑠subscript𝑧subscript𝑦𝐶𝑑𝑦𝑧r+s=h_{\ell}(z)-h_{\ell}(y)\leq Cd(y,z) by (9.7). In particular, |tu|=r+sCC4d(y,y)𝑡𝑢𝑟𝑠𝐶subscript𝐶4𝑑𝑦superscript𝑦|t-u|=r+s\leq CC_{4}d(y,y^{\prime}). Also 0rr+sCdiamYinfh0𝑟𝑟𝑠𝐶diam𝑌infimum0\leq r\leq r+s\leq C\operatorname{diam}Y\leq\inf h. Hence by (9.3) and (9.5),

d(Tuy,Ttz)=d(Trfy,Trfz)C5d(fy,fz)C2C5d(y,z)C2C4C5d(y,y).𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧𝑑subscript𝑇𝑟superscript𝑓𝑦subscript𝑇𝑟superscript𝑓𝑧subscript𝐶5𝑑superscript𝑓𝑦superscript𝑓𝑧subscript𝐶2subscript𝐶5𝑑𝑦𝑧subscript𝐶2subscript𝐶4subscript𝐶5𝑑𝑦superscript𝑦d(T_{u}y,T_{t}z)=d(T_{r}f^{\ell}y,T_{r}f^{\ell}z)\leq C_{5}d(f^{\ell}y,f^{\ell}z)\leq C_{2}C_{5}d(y,z)\leq C_{2}C_{4}C_{5}d(y,y^{\prime}).

The argument for =+1superscript1\ell^{\prime}=\ell+1 is analogous.

Choosing tsuperscript𝑡t^{\prime}. This goes along similar lines. We can shrink diamYdiam𝑌\operatorname{diam}Y and increase n0subscript𝑛0n_{0} so that C(C2+1)(diamY+γn0)infh𝐶subscript𝐶21diam𝑌superscript𝛾subscript𝑛0infimumC(C_{2}+1)(\operatorname{diam}Y+\gamma^{n_{0}})\leq\inf h. Note that s(z,y)=s(y,y)n01𝑠𝑧superscript𝑦𝑠𝑦superscript𝑦subscript𝑛01s(z,y^{\prime})=s(y,y^{\prime})\geq n_{0}\geq 1.

Since s(z,y)1𝑠𝑧superscript𝑦1s(z,y^{\prime})\geq 1, it follows from (7.4) that FzWu(Fy)𝐹𝑧superscript𝑊𝑢𝐹superscript𝑦Fz\in W^{u}(Fy^{\prime}). Write Tuz=TrfFzsubscript𝑇𝑢𝑧subscript𝑇𝑟superscript𝑓𝐹𝑧T_{u}z=T_{-r}f^{-\ell}Fz where 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1 and r[0,h(f(+1)Fz))𝑟0superscript𝑓1𝐹𝑧r\in[0,h(f^{-(\ell+1)}Fz)). Similarly write Tuy=TrfFysubscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑟superscript𝑓superscript𝐹superscript𝑦T_{u}y^{\prime}=T_{-r^{\prime}}f^{-\ell^{\prime}}Fy^{\prime}. Note that u=hτ(y)(z)r=hτ(y)(y)r𝑢subscript𝜏𝑦𝑧𝑟subscript𝜏𝑦superscriptsuperscript𝑦superscript𝑟u=h_{\tau(y)-\ell}(z)-r=h_{\tau(y)-\ell^{\prime}}(y^{\prime})-r^{\prime}.

Again, we show that ||1superscript1|\ell-\ell^{\prime}|\leq 1. By (9.7),

(1)infhsuperscript1infimum\displaystyle(\ell-\ell^{\prime}-1)\inf h hτ(y)1(y)hτ(y)(y)absentsubscript𝜏𝑦superscript1superscript𝑦subscript𝜏𝑦superscript𝑦\displaystyle\leq h_{\tau(y)-\ell^{\prime}-1}(y^{\prime})-h_{\tau(y)-\ell}(y^{\prime})
hτ(y)1(y)hτ(y)(z)+C(diamY+γn0)absentsubscript𝜏𝑦superscript1superscript𝑦subscript𝜏𝑦𝑧𝐶diam𝑌superscript𝛾subscript𝑛0\displaystyle\leq h_{\tau(y)-\ell^{\prime}-1}(y^{\prime})-h_{\tau(y)-\ell}(z)+C(\operatorname{diam}Y+\gamma^{n_{0}})
=rrh(fτ(y)1y)+C(diamY+γn0)C(diamY+γn0)infh.absentsuperscript𝑟𝑟superscript𝑓𝜏𝑦superscript1superscript𝑦𝐶diam𝑌superscript𝛾subscript𝑛0𝐶diam𝑌superscript𝛾subscript𝑛0infimum\displaystyle=r^{\prime}-r-h(f^{\tau(y)-\ell^{\prime}-1}y^{\prime})+C(\operatorname{diam}Y+\gamma^{n_{0}})\leq C(\operatorname{diam}Y+\gamma^{n_{0}})\leq\inf h.

Hence +1superscript1\ell\leq\ell^{\prime}+1. Similarly, (1)infhhτ(y)1(z)hτ(y)(z)infhsuperscript1infimumsubscript𝜏𝑦1𝑧subscript𝜏𝑦superscript𝑧infimum(\ell^{\prime}-\ell-1)\inf h\leq h_{\tau(y)-\ell-1}(z)-h_{\tau(y)-\ell^{\prime}}(z)\leq\inf h so ||1superscript1|\ell-\ell^{\prime}|\leq 1.

If =superscript\ell=\ell^{\prime}, then we take t=usuperscript𝑡𝑢t^{\prime}=u. It follows from (7.3) and (9.7) that

|rr|=|hτ(y)(y)hτ(y)(z)|C(d(y,z)+γs(y,z))C(C2+1)γs(y,z).𝑟superscript𝑟subscript𝜏𝑦superscript𝑦subscript𝜏𝑦𝑧𝐶𝑑superscript𝑦𝑧superscript𝛾𝑠superscript𝑦𝑧𝐶subscript𝐶21superscript𝛾𝑠superscript𝑦𝑧|r-r^{\prime}|=|h_{\tau(y)-\ell}(y^{\prime})-h_{\tau(y)-\ell}(z)|\leq C(d(y^{\prime},z)+\gamma^{s(y^{\prime},z)})\leq C(C_{2}+1)\gamma^{s(y^{\prime},z)}.

Also, by (7.3) and (9.4),

d(fFy,fFz)C2d(Fy,Fz)C22γ1γs(y,z).𝑑superscript𝑓𝐹superscript𝑦superscript𝑓𝐹𝑧subscript𝐶2𝑑𝐹superscript𝑦𝐹𝑧superscriptsubscript𝐶22superscript𝛾1superscript𝛾𝑠superscript𝑦𝑧d(f^{-\ell}Fy^{\prime},f^{-\ell}Fz)\leq C_{2}d(Fy^{\prime},Fz)\leq C_{2}^{2}\gamma^{-1}\gamma^{s(y^{\prime},z)}.

Without loss, rrsuperscript𝑟𝑟r^{\prime}\leq r, so by (7.3), (9.2) and (9.6),

d(Tuy,Tuz)𝑑subscript𝑇𝑢superscript𝑦subscript𝑇𝑢𝑧\displaystyle d(T_{u}y^{\prime},T_{u}z) =d(TrfFy,TrfFz)absent𝑑subscript𝑇superscript𝑟superscript𝑓𝐹superscript𝑦subscript𝑇𝑟superscript𝑓𝐹𝑧\displaystyle=d(T_{-r^{\prime}}f^{-\ell}Fy^{\prime},T_{-r}f^{-\ell}Fz)
d(TrfFy,TrfFz)+d(TrfFz,TrfFz)absent𝑑subscript𝑇superscript𝑟superscript𝑓𝐹superscript𝑦subscript𝑇superscript𝑟superscript𝑓𝐹𝑧𝑑subscript𝑇superscript𝑟superscript𝑓𝐹𝑧subscript𝑇𝑟superscript𝑓𝐹𝑧\displaystyle\leq d(T_{-r^{\prime}}f^{-\ell}Fy^{\prime},T_{-r^{\prime}}f^{-\ell}Fz)+d(T_{-r^{\prime}}f^{-\ell}Fz,T_{-r}f^{-\ell}Fz)
C5d(fFy,fFz)+|rr|γs(y,z)=γs(y,y).absentsubscript𝐶5𝑑superscript𝑓𝐹superscript𝑦superscript𝑓𝐹𝑧𝑟superscript𝑟much-less-thansuperscript𝛾𝑠superscript𝑦𝑧superscript𝛾𝑠𝑦superscript𝑦\displaystyle\leq C_{5}d(f^{-\ell}Fy^{\prime},f^{-\ell}Fz)+|r-r^{\prime}|\ll\gamma^{s(y^{\prime},z)}=\gamma^{s(y,y^{\prime})}.

If =1superscript1\ell=\ell^{\prime}-1, then we take t=urssuperscript𝑡𝑢superscript𝑟𝑠t^{\prime}=u-r^{\prime}-s where s=h(f(1)Fz)r0𝑠superscript𝑓1𝐹𝑧𝑟0s=h(f^{-(\ell-1)}Fz)-r\geq 0. Then Tuy=TrfFysubscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑟superscript𝑓superscript𝐹superscript𝑦T_{u}y^{\prime}=T_{-r^{\prime}}f^{-\ell^{\prime}}Fy^{\prime} and Ttz=TrsTuz=TrfFzsubscript𝑇superscript𝑡𝑧subscript𝑇superscript𝑟𝑠subscript𝑇𝑢𝑧subscript𝑇superscript𝑟superscript𝑓superscript𝐹𝑧T_{t^{\prime}}z=T_{-r^{\prime}-s}T_{u}z=T_{-r^{\prime}}f^{-\ell^{\prime}}Fz.

Note that u=hτ(y)(y)r=hτ(y)(z)+s𝑢subscript𝜏𝑦superscriptsuperscript𝑦superscript𝑟subscript𝜏𝑦superscript𝑧𝑠u=h_{\tau(y)-\ell^{\prime}}(y^{\prime})-r^{\prime}=h_{\tau(y)-\ell^{\prime}}(z)+s, hence r+s=hτ(y)(y)hτ(y)(z)C(C2+1)γs(y,y)superscript𝑟𝑠subscript𝜏𝑦superscriptsuperscript𝑦subscript𝜏𝑦superscript𝑧𝐶subscript𝐶21superscript𝛾𝑠𝑦superscript𝑦r^{\prime}+s=h_{\tau(y)-\ell^{\prime}}(y^{\prime})-h_{\tau(y)-\ell^{\prime}}(z)\leq C(C_{2}+1)\gamma^{s(y,y^{\prime})} by (9.7). In particular, |tu|=r+sγs(y,y)superscript𝑡𝑢superscript𝑟𝑠much-less-thansuperscript𝛾𝑠𝑦superscript𝑦|t^{\prime}-u|=r^{\prime}+s\ll\gamma^{s(y,y^{\prime})}. Also, 0rr+sC(C2+1)γn0infh0superscript𝑟superscript𝑟𝑠𝐶subscript𝐶21superscript𝛾subscript𝑛0infimum0\leq r^{\prime}\leq r^{\prime}+s\leq C(C_{2}+1)\gamma^{n_{0}}\leq\inf h. Hence by (7.3), (9.4) and (9.6),

d(Tuy,Ttz)=d(TrfFy,TrfFz)C5d(fFy,fFz)γs(y,y).𝑑subscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑡𝑧𝑑subscript𝑇superscript𝑟superscript𝑓superscript𝐹superscript𝑦subscript𝑇superscript𝑟superscript𝑓superscript𝐹𝑧subscript𝐶5𝑑superscript𝑓superscript𝐹superscript𝑦superscript𝑓superscript𝐹𝑧much-less-thansuperscript𝛾𝑠𝑦superscript𝑦d(T_{u}y^{\prime},T_{t^{\prime}}z)=d(T_{-r^{\prime}}f^{-\ell^{\prime}}Fy^{\prime},T_{-r^{\prime}}f^{-\ell^{\prime}}Fz)\leq C_{5}d(f^{-\ell^{\prime}}Fy^{\prime},f^{-\ell^{\prime}}Fz)\ll\gamma^{s(y,y^{\prime})}.

The argument for =+1superscript1\ell=\ell^{\prime}+1 is analogous. ∎

Corollary 9.6

Let vC0,η(M)𝑣superscript𝐶0𝜂𝑀v\in C^{0,\eta}(M), wC0,m(M)𝑤superscript𝐶0𝑚𝑀w\in C^{0,m}(M) such that tkwC0,η(M)superscriptsubscript𝑡𝑘𝑤superscript𝐶0𝜂𝑀\partial_{t}^{k}w\in C^{0,\eta}(M), for all k=0,,m𝑘0𝑚k=0,\dots,m. Suppose also that there is a constant C>0𝐶0C>0 such that |v(x)v(x)|Cd(x,x)η𝑣𝑥𝑣superscript𝑥𝐶𝑑superscript𝑥superscript𝑥𝜂|v(x)-v(x^{\prime})|\leq Cd(x,x^{\prime})^{\eta} and |tkw(x)tkw(x)|Cd(x,x)ηsuperscriptsubscript𝑡𝑘𝑤𝑥superscriptsubscript𝑡𝑘𝑤superscript𝑥𝐶𝑑superscript𝑥superscript𝑥𝜂|\partial_{t}^{k}w(x)-\partial_{t}^{k}w(x^{\prime})|\leq Cd(x,x^{\prime})^{\eta} for all x,xM𝑥superscript𝑥𝑀x,x^{\prime}\in M of the form x=Tuy𝑥subscript𝑇𝑢𝑦x=T_{u}y, x=Tuysuperscript𝑥subscript𝑇𝑢superscript𝑦x^{\prime}=T_{u}y^{\prime} where y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j} for some j1𝑗1j\geq 1, u[0,φ(y)]𝑢0𝜑𝑦u\in[0,\varphi(y)], u[0,φ(y)]superscript𝑢0𝜑superscript𝑦u^{\prime}\in[0,\varphi(y^{\prime})], and for all k=0,,m𝑘0𝑚k=0,\dots,m. Then hh, Ttsubscript𝑇𝑡T_{t}, v𝑣v and w𝑤w are dynamically Hölder in the sense of Definition 7.6.

Proof.

Condition (a) of Definition 7.6 follows from Proposition 9.3. To check condition (b), we distinguish two cases. If s(y,y)<n0𝑠𝑦superscript𝑦subscript𝑛0s(y,y^{\prime})<n_{0}, we may take t=t=u𝑡superscript𝑡𝑢t=t^{\prime}=u and use that |v(x)v(x)|2|v|γn0𝑣𝑥𝑣superscript𝑥2subscript𝑣much-less-thansuperscript𝛾subscript𝑛0|v(x)-v(x^{\prime})|\leq 2|v|_{\infty}\ll\gamma^{n_{0}} for any x,xM𝑥superscript𝑥𝑀x,x^{\prime}\in M. If s(y,y)n0𝑠𝑦superscript𝑦subscript𝑛0s(y,y^{\prime})\geq n_{0}, Proposition 9.5 applies and, along with Formulas (9.1)–(9.6), implies Definition 7.6(b) . ∎

9.2 Tail estimate for φ𝜑\varphi and completion of the proof of Theorem 9.1

Since

μX(xX:h(x)>t)\displaystyle\mu_{X}(x\in X:h(x)>t) =O(t2)absent𝑂superscript𝑡2\displaystyle=O(t^{-2}) (9.8)
μ(yY:τ(y)>n)\displaystyle\mu(y\in Y:\tau(y)>n) =O(ecn)for some c>0,absent𝑂superscript𝑒𝑐𝑛for some c>0\displaystyle=O(e^{-cn})\quad\mbox{for some $c>0$}, (9.9)

a standard argument shows that μ(φ>t)=O((logt)2t2)𝜇𝜑𝑡𝑂superscript𝑡2superscript𝑡2\mu(\varphi>t)=O((\log t)^{2}t^{-2}). In fact, we have

Proposition 9.7

μ(φ>t)=O(t2)𝜇𝜑𝑡𝑂superscript𝑡2\mu(\varphi>t)=O(t^{-2}).

The crucial ingredient for proving Proposition 9.7 is due to Szász & Varjú [29].

Lemma 9.8 ( [29, Lemma 16], [15, Lemma 5.1] )

There are constants p,q>0𝑝𝑞0p,q>0 with the following property. Define

Xb(m)={xX:[h(x)]=mandh(Tjx)>m1qfor some j{1,,blogm}}.subscript𝑋𝑏𝑚conditional-set𝑥𝑋delimited-[]𝑥𝑚andsuperscript𝑇𝑗𝑥superscript𝑚1𝑞for some j{1,,blogm}X_{b}(m)=\big{\{}x\in X:[h(x)]=m\enspace\text{and}\enspace h(T^{j}x)>m^{1-q}\enspace\text{for some $j\in\{1,\dots,b\log m\}$}\big{\}}.

Then for any b𝑏b sufficiently large there is a constant C=C(b)>0𝐶𝐶𝑏0C=C(b)>0 such that

μX(Xb(m))CmpμX(xX:[h(x)]=m)for all m1.\mu_{X}(X_{b}(m))\leq Cm^{-p}\mu_{X}(x\in X:[h(x)]=m)\quad\text{for all $m\geq 1$.}

For b>0𝑏0b>0, define

Yb(n)={yY:τ(y)blognandmax0<τ(y)h(Ty)12nandφ(y)n}.subscript𝑌𝑏𝑛conditional-set𝑦𝑌𝜏𝑦𝑏𝑛andsubscript0𝜏𝑦superscript𝑇𝑦12𝑛and𝜑𝑦𝑛Y_{b}(n)=\{y\in Y:\tau(y)\leq b\log n\enspace\text{and}\enspace\max_{0\leq\ell<\tau(y)}h(T^{\ell}y)\leq{\textstyle\frac{1}{2}n}\enspace\text{and}\enspace\varphi(y)\geq n\}.
Corollary 9.9

For b𝑏b sufficiently large, μ(Yb(n))=o(n2)𝜇subscript𝑌𝑏𝑛𝑜superscript𝑛2\mu(Y_{b}(n))=o(n^{-2}).

Proof.

Fix p𝑝p and q𝑞q as in Lemma 9.8. Also fix b𝑏b sufficiently large.

Let yYb(n)𝑦subscript𝑌𝑏𝑛y\in Y_{b}(n). Define h1(y)=max0<τ(y)h(fy)subscript1𝑦subscript0𝜏𝑦superscript𝑓𝑦h_{1}(y)=\max_{0\leq\ell<\tau(y)}h(f^{\ell}y) and choose 1(y){0,,τ(y)1}subscript1𝑦0𝜏𝑦1\ell_{1}(y)\in\{0,\dots,\tau(y)-1\} such that h1(y)=h(f1(y)y)subscript1𝑦superscript𝑓subscript1𝑦𝑦h_{1}(y)=h(f^{\ell_{1}(y)}y). Define h2(y)=max0<τ(y),1(y)h(fy)subscript2𝑦subscriptformulae-sequence0𝜏𝑦subscript1𝑦superscript𝑓𝑦h_{2}(y)=\max_{0\leq\ell<\tau(y),\,\ell\neq\ell_{1}(y)}h(f^{\ell}y). Then h1(y)subscript1𝑦h_{1}(y) and h2(y)subscript2𝑦h_{2}(y) are the two largest free flights hfsuperscript𝑓h\circ f^{\ell} during the iterates =0,,τ(y)10𝜏𝑦1\ell=0,\dots,\tau(y)-1.

We begin by showing that these two flight times have comparable length. Indeed, let mi=[hi]subscript𝑚𝑖delimited-[]subscript𝑖m_{i}=[h_{i}], i=1,2𝑖12i=1,2. Then nφh1+(τ1)h2n/2+(blogn)h2𝑛𝜑subscript1𝜏1subscript2𝑛2𝑏𝑛subscript2n\leq\varphi\leq h_{1}+(\tau-1)h_{2}\leq n/2+(b\log n)h_{2}. Hence

n2blogn1m2m1n2.𝑛2𝑏𝑛1subscript𝑚2subscript𝑚1𝑛2\frac{n}{2b\log n}-1\leq m_{2}\leq m_{1}\leq\frac{n}{2}. (9.10)

In particular, m1>m21qsubscript𝑚1superscriptsubscript𝑚21𝑞m_{1}>m_{2}^{1-q} and m2>m11qsubscript𝑚2superscriptsubscript𝑚11𝑞m_{2}>m_{1}^{1-q} for large n𝑛n.

Choose 2(y){0,,τ(y)1}subscript2𝑦0𝜏𝑦1\ell_{2}(y)\in\{0,\dots,\tau(y)-1\} such that 2(y)1(y)subscript2𝑦subscript1𝑦\ell_{2}(y)\neq\ell_{1}(y) and h2(y)=h(f2(y)y)subscript2𝑦superscript𝑓subscript2𝑦𝑦h_{2}(y)=h(f^{\ell_{2}(y)}y). We can suppose without loss that 1(y)<2(y)subscript1𝑦subscript2𝑦\ell_{1}(y)<\ell_{2}(y). For large n𝑛n, it follows from (9.10) that f1(y)yXb(m1(y))superscript𝑓subscript1𝑦𝑦subscript𝑋𝑏subscript𝑚1𝑦f^{\ell_{1}(y)}y\in X_{b}(m_{1}(y)). Hence

Yb(n)fXb(m)for some <blognmn/(2blogn)1,subscript𝑌𝑏𝑛superscript𝑓subscript𝑋𝑏𝑚for some <blognmn/(2blogn)1,Y_{b}(n)\subset f^{-\ell}X_{b}(m)\quad\text{for some $\ell<b\log n$, $m\geq n/(2b\log n)-1$,}

and so

μ(Yb(n))μX(Yb(n)×0)blognmn/(2blogn)1μX(Xb(m)).much-less-than𝜇subscript𝑌𝑏𝑛subscript𝜇𝑋subscript𝑌𝑏𝑛0𝑏𝑛subscript𝑚𝑛2𝑏𝑛1subscript𝜇𝑋subscript𝑋𝑏𝑚\mu(Y_{b}(n))\ll\mu_{X}(Y_{b}(n)\times 0)\leq b\log n\sum_{m\geq n/(2b\log n)-1}\mu_{X}(X_{b}(m)).

By Lemma 9.8 and (9.8),

μ(Yb(n))𝜇subscript𝑌𝑏𝑛\displaystyle\mu(Y_{b}(n)) lognmn/(2blogn)1mpμX(xX:[h(x)]=m)\displaystyle\ll\log n\sum_{m\geq n/(2b\log n)-1}m^{-p}\mu_{X}(x\in X:[h(x)]=m)
logn(n/logn)(2+p)=o(n2),much-less-thanabsent𝑛superscript𝑛𝑛2𝑝𝑜superscript𝑛2\displaystyle\ll\log n(n/\log n)^{-(2+p)}=o(n^{-2}),

as required. ∎

Proof of Proposition 9.7   Define the tower Δ={(y,)Y×:0τ(y)1}Δconditional-set𝑦𝑌0𝜏𝑦1\Delta=\{(y,\ell)\in Y\times{\mathbb{Z}}:0\leq\ell\leq\tau(y)-1\} with probability measure μΔ=μ×counting/τ¯subscript𝜇Δ𝜇counting¯𝜏\mu_{\Delta}=\mu\times{\rm counting}/\bar{\tau} where τ¯=Yτ𝑑μ¯𝜏subscript𝑌𝜏differential-d𝜇\bar{\tau}=\int_{Y}\tau\,d\mu. Recall that μX=πμΔsubscript𝜇𝑋subscript𝜋subscript𝜇Δ\mu_{X}=\pi_{*}\mu_{\Delta} where π(y,)=fy𝜋𝑦superscript𝑓𝑦\pi(y,\ell)=f^{\ell}y.

Write max0<τ(y)h(fy)=h(f1(y)y)subscript0𝜏𝑦superscript𝑓𝑦superscript𝑓subscript1𝑦𝑦\max_{0\leq\ell<\tau(y)}h(f^{\ell}y)=h(f^{\ell_{1}(y)}y) where 1(y){0,,τ(y)1}subscript1𝑦0𝜏𝑦1\ell_{1}(y)\in\{0,\dots,\tau(y)-1\}. Then

μ{y\displaystyle\mu\{y\in Y:max0<τ(y)h(fy)>n/2}=τ¯μΔ{(y,0)Δ:h(f1(y)y)>n/2}\displaystyle Y:\max_{0\leq\ell<\tau(y)}h(f^{\ell}y)>n/2\}=\bar{\tau}\mu_{\Delta}\{(y,0)\in\Delta:h(f^{\ell_{1}(y)}y)>n/2\}
=τ¯μΔ{(y,1(y)):h(f1(y)y)>n/2}=τ¯μΔ{(y,1(y)):hπ(y,1(y))>n/2}absent¯𝜏subscript𝜇Δconditional-set𝑦subscript1𝑦superscript𝑓subscript1𝑦𝑦𝑛2¯𝜏subscript𝜇Δconditional-set𝑦subscript1𝑦𝜋𝑦subscript1𝑦𝑛2\displaystyle=\bar{\tau}\mu_{\Delta}\{(y,\ell_{1}(y)):h(f^{\ell_{1}(y)}y)>n/2\}=\bar{\tau}\mu_{\Delta}\{(y,\ell_{1}(y)):h\circ\pi(y,\ell_{1}(y))>n/2\}
τ¯μΔ{pΔ:hπ(p)>n/2}=τ¯μX{xX:h(x)>n/2},absent¯𝜏subscript𝜇Δconditional-set𝑝Δ𝜋𝑝𝑛2¯𝜏subscript𝜇𝑋conditional-set𝑥𝑋𝑥𝑛2\displaystyle\leq\bar{\tau}\mu_{\Delta}\{p\in\Delta:h\circ\pi(p)>n/2\}=\bar{\tau}\mu_{X}\{x\in X:h(x)>n/2\},

and so μ{yY:max0<τ(y)h(Ty)>n/2}=O(n2)𝜇conditional-set𝑦𝑌subscript0𝜏𝑦superscript𝑇𝑦𝑛2𝑂superscript𝑛2\mu\{y\in Y:\max_{0\leq\ell<\tau(y)}h(T^{\ell}y)>n/2\}=O(n^{-2}) by (9.8). Hence it follows from Corollary 9.9 that

μ{yY:τ(y)blognandφ(y)n}=O(n2).𝜇conditional-set𝑦𝑌𝜏𝑦𝑏𝑛and𝜑𝑦𝑛𝑂superscript𝑛2\mu\{y\in Y:\tau(y)\leq b\log n\enspace\text{and}\enspace\varphi(y)\geq n\}=O(n^{-2}).

Finally, by (9.9), μ(τ>blogn)=O(nbc)=o(n2)𝜇𝜏𝑏𝑛𝑂superscript𝑛𝑏𝑐𝑜superscript𝑛2\mu(\tau>b\log n)=O(n^{-bc})=o(n^{-2}) for any b>2/c𝑏2𝑐b>2/c and so μ(φn)=O(n2)𝜇𝜑𝑛𝑂superscript𝑛2\mu(\varphi\geq n)=O(n^{-2}) as required. ∎

It follows from Lemma 8.3 and Corollary 9.4 that condition (H) is satisfied. Hence by Corollary 8.1(a), the suspension flow Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow as defined in Section 6. By Proposition 9.7, μ(φ>t)=O(t2)𝜇𝜑𝑡𝑂superscript𝑡2\mu(\varphi>t)=O(t^{-2}). By Corollary 9.6, the flows and observables are dynamically Hölder (Definition 7.6). Hence it follows from Corollary 8.1(b) that absence of approximate eigenfunctions implies decay rate O(t1)𝑂superscript𝑡1O(t^{-1}).

Finally, we exclude approximate eigenfunctions. By Corollary 9.4, condition (8.2) holds and hence the temporal distortion function D:Y×Y:𝐷𝑌𝑌D:Y\times Y\to{\mathbb{R}} is defined as in Section 8.4. Let Z¯0Y¯subscript¯𝑍0¯𝑌{\mkern 1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu}_{0}\subset{\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu} be a finite subsystem and let Z0=π¯1Z¯0subscript𝑍0superscript¯𝜋1subscript¯𝑍0Z_{0}=\bar{\pi}^{-1}{\mkern 1.5mu\overline{\mkern-1.5muZ\mkern-1.5mu}\mkern 1.5mu}_{0}. The presence of a contact structure implies by Remark 8.10 that the lower box dimension of D(Z0×Z0)𝐷subscript𝑍0subscript𝑍0D(Z_{0}\times Z_{0}) is positive. Hence absence of approximate eigenfunctions follows from Lemma 8.9.

9.3 Semi-dispersing Lorentz flows and stadia

In this subsection we discuss two further classes of billiard flows and show that the scheme presented above can be adapted to cover these examples, resulting in Theorem 9.13.

Semi-dispersing Lorentz flows are billiard flows in the planar domain obtained as RSk𝑅subscript𝑆𝑘R\setminus\bigcup S_{k} where R𝑅R is a rectangle and the SkRsubscript𝑆𝑘𝑅S_{k}\subset R are finitely many disjoint convex scatterers with C3superscript𝐶3C^{3} boundaries of nonvanishing curvature. By the unfolding process – tiling the plane with identical copies of R𝑅R, and reflecting the scatterers Sksubscript𝑆𝑘S_{k} across the sides of all these rectangles – an infinite periodic configuration is obtained, which can be regarded as an infinite horizon Lorentz gas.

Bunimovich stadia are convex billiard domains enclosed by two semicircular arcs (of equal radii) connected by two parallel line segments. An unfolding process could reduce the bounces on the parallel line segments to long flights in an unbounded domain, however, there is another quasi-integrable effect here corresponding to sequences of consecutive collisions on the same semi-circular arc.

Both of these examples have been extensively studied in the literature, see for instance [9, 13, 14, 25, 6], and references therein. A common feature of the two examples is that the billiard map itself is not uniformly hyperbolic; however, there is a geometrically defined first return map which has uniform expansion rates. As before, the billiard domain is denoted by Q𝑄Q, and the billiard flow is Tt:MM:subscript𝑇𝑡𝑀𝑀T_{t}:M\to M where M=Q×𝕊1𝑀𝑄superscript𝕊1M=Q\times{\mathbb{S}}^{1}. However, this time we prefer to denote the natural Poincaré section Q×[π/2,π/2]M𝑄𝜋2𝜋2𝑀\partial Q\times[-\pi/2,\pi/2]\subset M by X~~𝑋\widetilde{X}, the corresponding billiard map as f~:X~X~:~𝑓~𝑋~𝑋\tilde{f}:\widetilde{X}\to\widetilde{X}, and the free flight function as h~:X~+:~~𝑋superscript\tilde{h}:\widetilde{X}\to{\mathbb{R}}^{+} where h~(x~)=inf{t>0:Ttx~X~}~~𝑥infimumconditional-set𝑡0subscript𝑇𝑡~𝑥~𝑋\tilde{h}(\tilde{x})=\inf\{t>0:T_{t}\tilde{x}\in\widetilde{X}\}. Then, as mentioned above, there is a subset XX~𝑋~𝑋X\subset\widetilde{X} such that the first return map of f~~𝑓\tilde{f} to X𝑋X has good hyperbolic properties. We denote this first return map by f:XX:𝑓𝑋𝑋f:X\to X. The corresponding free flight function h:X+:𝑋superscripth:X\to{\mathbb{R}}^{+} is given by h(x)=inf{t>0:TtxX}𝑥infimumconditional-set𝑡0subscript𝑇𝑡𝑥𝑋h(x)=\inf\{t>0:T_{t}x\in X\}. Let us, furthermore, introduce the discrete return time r~:X+:~𝑟𝑋superscript\tilde{r}:X\to{\mathbb{Z}}^{+} given by r~(x)=min{n1:f~nxX}~𝑟𝑥:𝑛1superscript~𝑓𝑛𝑥𝑋\tilde{r}(x)=\min\{n\geq 1:\tilde{f}^{n}x\in X\}.

In the case of the semi-dispersing Lorentz flow, X𝑋X corresponds to collisions on the scatterers Sksubscript𝑆𝑘S_{k}. In the case of the stadium, X𝑋X corresponds to first bounces on semi-circular arcs, that is, xX𝑥𝑋x\in X if x𝑥x is on one of the semi-circular arcs, but f~1xsuperscript~𝑓1𝑥\tilde{f}^{-1}x is on another boundary component (on the other semi-circular arc, or on one of the line segments).

The following properties hold. Unless otherwise stated, standard references are [13, Chapter 8] and [14]. As in section 9.1, d(x,x)𝑑𝑥superscript𝑥d(x,x^{\prime}) always denotes the Euclidean distance of the two points, generated by the Riemannian metric.

  • There is a countable partition X𝒮=m=1Xm𝑋𝒮superscriptsubscript𝑚1subscript𝑋𝑚X\setminus{\mathcal{S}}=\bigcup_{m=1}^{\infty}X_{m} such that f|Xmevaluated-at𝑓subscript𝑋𝑚f|_{X_{m}} is C2superscript𝐶2C^{2} and r~|Xmevaluated-at~𝑟subscript𝑋𝑚\tilde{r}|_{X_{m}} is constant for any m1𝑚1m\geq 1. We refer to the partition elements Xmsubscript𝑋𝑚X_{m} with r~|Xm2evaluated-at~𝑟subscript𝑋𝑚2\tilde{r}|_{X_{m}}\geq 2 as cells; these are of two different types:

    • Bouncing cells are present both in the semi-dispersing billiard examples and in stadia. For these, one iteration of f|Xmevaluated-at𝑓subscript𝑋𝑚f|_{X_{m}} consists of several consecutive reflections on the flat boundary components, that is, the line segments. By the above mentioned unfolding process, these reflections reduce to trajectories along straight lines in the associated unbounded table.

    • Sliding cells are present only in stadia. For these, one iteration of f|Xmevaluated-at𝑓subscript𝑋𝑚f|_{X_{m}} consists of several consecutive collisions on the same semi-circular arc.

  • infh>0infimum0\inf h>0, and suph~<supremum~\sup\tilde{h}<\infty, however, there is no uniform upper bound on hh, and no uniform lower bound for h~~\tilde{h}.

  • f:XX:𝑓𝑋𝑋f:X\to X is uniformly hyperbolic in the sense that stable and unstable manifolds exist for almost every x𝑥x, and Formulas (9.3) and (9.4) hold. This follows from the uniform expansion rates of f𝑓f, see [13, Formula (8.22)].

  • If x,xXm𝑥superscript𝑥subscript𝑋𝑚x,x^{\prime}\in X_{m} where Xmsubscript𝑋𝑚X_{m} is a bouncing cell, in the associated unfolded table the flow trajectories until the first return to X𝑋X are straight lines, hence (9.1) follows. If x,xXm𝑥superscript𝑥subscript𝑋𝑚x,x^{\prime}\in X_{m} and Xmsubscript𝑋𝑚X_{m} is a sliding cell, the induced roof function is uniformly Hölder continuous with exponent 1/4141/4, as established in the proof of [6, Theorem 3.1]. The same geometric reasoning applies to h~k(x)=h~(x)+h~(f~x)++h~(f~k1x)subscript~𝑘𝑥~𝑥~~𝑓𝑥~superscript~𝑓𝑘1𝑥\tilde{h}_{k}(x)=\tilde{h}(x)+\tilde{h}(\tilde{f}x)+\dots+\tilde{h}(\tilde{f}^{k-1}x) as long as kr~(x)𝑘~𝑟𝑥k\leq\tilde{r}(x). Summarizing, we have

    |h~k(x)h~k(x)|d(x,x)1/4+d(fx,fx)1/4much-less-thansubscript~𝑘𝑥subscript~𝑘superscript𝑥𝑑superscript𝑥superscript𝑥14𝑑superscript𝑓𝑥𝑓superscript𝑥14|\tilde{h}_{k}(x)-\tilde{h}_{k}(x^{\prime})|\ll d(x,x^{\prime})^{1/4}+d(fx,fx^{\prime})^{1/4} (9.11)

    for x,xXm,m1formulae-sequence𝑥superscript𝑥subscript𝑋𝑚𝑚1x,x^{\prime}\in X_{m},\ m\geq 1 and kr~(x)1𝑘~𝑟𝑥1k\leq\tilde{r}(x)-1. In particular, |h(x)h(x)|d(x,x)1/4+d(fx,fx)1/4much-less-than𝑥superscript𝑥𝑑superscript𝑥superscript𝑥14𝑑superscript𝑓𝑥𝑓superscript𝑥14|h(x)-h(x^{\prime})|\ll d(x,x^{\prime})^{1/4}+d(fx,fx^{\prime})^{1/4}.

  • (9.2) has to be relaxed to

    d(Ttx~,Ttx~)|tt|for allx~X~and t,t[0,h~(x~)).formulae-sequence𝑑subscript𝑇𝑡~𝑥subscript𝑇superscript𝑡~𝑥𝑡superscript𝑡formulae-sequencefor all~𝑥~𝑋and 𝑡superscript𝑡0~~𝑥d(T_{t}\tilde{x},T_{t^{\prime}}\tilde{x})\leq|t-t^{\prime}|\quad\text{for\ all}\ \tilde{x}\in\widetilde{X}\ \text{and\ }t,t^{\prime}\in[0,\tilde{h}(\tilde{x})). (9.12)
  • (9.5) has to be relaxed to the following two formulas:

    d(Ttx~,Ttx~)d(x~,x~)formuch-less-than𝑑subscript𝑇𝑡~𝑥subscript𝑇𝑡superscript~𝑥𝑑~𝑥superscript~𝑥for\displaystyle d(T_{t}\tilde{x},T_{t}\tilde{x}^{\prime})\ll d(\tilde{x},\tilde{x}^{\prime})\quad\text{for}\ x~X~,x~Ws(x~),t[0,h~(x~))[0,h~(x~));formulae-sequence~𝑥~𝑋formulae-sequencesuperscript~𝑥superscript𝑊𝑠~𝑥𝑡0~~𝑥0~superscript~𝑥\displaystyle\tilde{x}\in\widetilde{X},\tilde{x}^{\prime}\in W^{s}(\tilde{x}),\ t\in[0,\tilde{h}(\tilde{x}))\cap[0,\tilde{h}(\tilde{x}^{\prime})); (9.13)
    d(f~kx,f~kx)d(x,x)formuch-less-than𝑑superscript~𝑓𝑘𝑥superscript~𝑓𝑘superscript𝑥𝑑𝑥superscript𝑥for\displaystyle d(\tilde{f}^{k}x,\tilde{f}^{k}x^{\prime})\ll d(x,x^{\prime})\quad\text{for}\ xX,xWs(x), 0k.formulae-sequence𝑥𝑋formulae-sequencesuperscript𝑥superscript𝑊𝑠𝑥 0𝑘\displaystyle x\in X,x^{\prime}\in W^{s}(x),\ 0\leq k. (9.14)

    Similarly, (9.6) has to be relaxed to

    d(Ttx~,Ttx~)d(x~,x~)formuch-less-than𝑑subscript𝑇𝑡~𝑥subscript𝑇𝑡superscript~𝑥𝑑~𝑥superscript~𝑥for\displaystyle d(T_{-t}\tilde{x},T_{-t}\tilde{x}^{\prime})\ll d(\tilde{x},\tilde{x}^{\prime})\quad\text{for}\ x~X~,x~Wu(x~),formulae-sequence~𝑥~𝑋superscript~𝑥superscript𝑊𝑢~𝑥\displaystyle\tilde{x}\in\widetilde{X},\tilde{x}^{\prime}\in W^{u}(\tilde{x}),
    t[0,h~(f~1x~))[0,h~(f~1x~));𝑡0~superscript~𝑓1~𝑥0~superscript~𝑓1superscript~𝑥\displaystyle t\in[0,\tilde{h}(\tilde{f}^{-1}\tilde{x}))\cap[0,\tilde{h}(\tilde{f}^{-1}\tilde{x}^{\prime})); (9.15)
    d(f~kx,f~kx)d(x,x)formuch-less-than𝑑superscript~𝑓𝑘𝑥superscript~𝑓𝑘superscript𝑥𝑑𝑥superscript𝑥for\displaystyle d(\tilde{f}^{-k}x,\tilde{f}^{-k}x^{\prime})\ll d(x,x^{\prime})\quad\text{for}\ xX,xWu(x), 0k.formulae-sequence𝑥𝑋formulae-sequencesuperscript𝑥superscript𝑊𝑢𝑥 0𝑘\displaystyle x\in X,x^{\prime}\in W^{u}(x),\ 0\leq k. (9.16)

    To verify (9.16), let us note first that d(x,x)𝑑𝑥superscript𝑥d(x,x^{\prime}) consists of a position and a velocity component, and in course of a free flight velocities do not change. Now the mechanism of hyperbolicity for stadia is defocusing, see, for instance, [13, Figure 8.1], which guarantees that for xWu(x)superscript𝑥superscript𝑊𝑢𝑥x^{\prime}\in W^{u}(x), the position component of d(x,x)𝑑𝑥superscript𝑥d(x,x^{\prime}) in course of the free flight is dominated by the position component at the end of the free flight. (9.14) holds for analogous reasons. To verify (9.3), by uniform hyperbolicity of f𝑓f (in particular Formula (9.4), see above), it is enough to consider how f~~𝑓\tilde{f} evolves unstable vectors between two consecutive applications of f𝑓f, ie. within a series of sliding or bouncing collisions. On the one hand, again by the defocusing mechanism, f~~𝑓\tilde{f} does not contract the p-length of unstable vectors, see [13, Section 8.2]. On the other hand, for an unstable vector, the ratio of the Euclidean and the p-length is 1+𝒱2/cosφ1superscript𝒱2𝜑\sqrt{1+\mathcal{V}^{2}}/\cos\varphi, where 𝒱𝒱\mathcal{V} is the slope of the unstable vector in the standard billiard coordinates, and φ𝜑\varphi is the collision angle, see [13, Formula (8.21)]. Now |𝒱|𝒱|\mathcal{V}| is uniformly bounded away from \infty, see Formula [13, Formula (8.18)], while cosφ𝜑\cos\varphi is constant in course of a sequence of consecutive sliding or bouncing collisions. (9.13) holds by an analogous argument.

  • The map f:XX:𝑓𝑋𝑋f:X\to X can be modeled by a Young tower with exponential tails. In particular, there exists a subset YX𝑌𝑋Y\subset X and an induced map F=fτ:YY:𝐹superscript𝑓𝜏𝑌𝑌F=f^{\tau}:Y\to Y that possesses the properties discussed in Section 7.1 including (7.4). The tails of the return time τ:Y+:𝜏𝑌superscript\tau:Y\to{\mathbb{Z}}^{+} are exponential, i.e. μ(τ>n)=O(ecn)𝜇𝜏𝑛𝑂superscript𝑒𝑐𝑛\mu(\tau>n)=O(e^{-cn}) for some c>0𝑐0c>0.444It is important to note that here τ𝜏\tau is the return time to Y𝑌Y in terms of f𝑓f; the return time in terms of f~~𝑓\tilde{f} has polynomial tails. Moreover, the construction can be carried out so that diamYdiam𝑌\operatorname{diam}Y is as small as desired. The existence of the Young tower satisfying these properties is established in [14]. As in subsection 9.1, we introduce the induced roof function φ==0τ1hf𝜑superscriptsubscript0𝜏1superscript𝑓\varphi=\sum_{\ell=0}^{\tau-1}h\circ f^{\ell}.

  • By construction, for y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j}, j1𝑗1j\geq 1 and τ𝜏\ell\leq\tau fixed, fysuperscript𝑓𝑦f^{\ell}y and fysuperscript𝑓superscript𝑦f^{\ell}y^{\prime} always belong to the same cell of X𝑋X.

Let us introduce γ^=γ1/4^𝛾superscript𝛾14\hat{\gamma}=\gamma^{1/4} and d¯(y,y)=d(y,y)1/4¯𝑑𝑦superscript𝑦𝑑superscript𝑦superscript𝑦14\bar{d}(y,y^{\prime})=d(y,y^{\prime})^{1/4}. The following version of Proposition 9.3 holds.

Proposition 9.10

For all y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j}, j1𝑗1j\geq 1, and all 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1,

|h(fy)h(fy)|γ^(d¯(y,y)+γ^τ(y)γ^s(y,y).|h(f^{\ell}y)-h(f^{\ell}y^{\prime})|\ll\hat{\gamma}^{\ell}(\bar{d}(y,y^{\prime})+\hat{\gamma}^{\tau(y)-\ell}\hat{\gamma}^{s(y,y^{\prime})}.
Proof.

The proof of Proposition 9.3 applies, using (9.11) instead of (9.1). ∎

This readily implies

Corollary 9.11

Conditions (8.1) and (8.2) hold, with γ𝛾\gamma replaced by γ^^𝛾\hat{\gamma}, and d(y,y)𝑑𝑦superscript𝑦d(y,y^{\prime}) replaced by d¯(y,y)¯𝑑𝑦superscript𝑦\bar{d}(y,y^{\prime}). ∎

The adapted version of Proposition 9.5 reads as follows.

Proposition 9.12

For diamYdiam𝑌\operatorname{diam}Y sufficiently small, there exist an integer n01subscript𝑛01n_{0}\geq 1 and a constant C>0𝐶0C>0 such that for all y,yY𝑦superscript𝑦𝑌y,y^{\prime}\in Y, s(y,y0)n0𝑠𝑦subscript𝑦0subscript𝑛0s(y,y_{0})\geq n_{0}, and all u[0,φ(y)][0,φ(y)]𝑢0𝜑𝑦0𝜑superscript𝑦u\in[0,\varphi(y)]\cap[0,\varphi(y^{\prime})], there exist t,t𝑡superscript𝑡t,t^{\prime}\in{\mathbb{R}} such that

|tu|𝑡𝑢\displaystyle|t-u| Cd¯(y,y),absent𝐶¯𝑑𝑦superscript𝑦\displaystyle\leq C\bar{d}(y,y^{\prime}), d(Tuy,Ttz)𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧\displaystyle\qquad d(T_{u}y,T_{t}z) Cd¯(y,y),absent𝐶¯𝑑𝑦superscript𝑦\displaystyle\leq C\bar{d}(y,y^{\prime}),
|tu|superscript𝑡𝑢\displaystyle|t^{\prime}-u| Cγ^s(y,y),absent𝐶superscript^𝛾𝑠𝑦superscript𝑦\displaystyle\leq C\hat{\gamma}^{s(y,y^{\prime})}, d(Tuy,Ttz)𝑑subscript𝑇𝑢superscript𝑦subscript𝑇superscript𝑡𝑧\displaystyle\qquad d(T_{u}y^{\prime},T_{t^{\prime}}z) Cγ^s(y,y),absent𝐶superscript^𝛾𝑠𝑦superscript𝑦\displaystyle\leq C\hat{\gamma}^{s(y,y^{\prime})},

where z=Ws(y)Wu(y)𝑧superscript𝑊𝑠𝑦superscript𝑊𝑢superscript𝑦z=W^{s}(y)\cap W^{u}(y^{\prime}).

Proof.

First, (9.7) can be updated as

|h(y)h(y)|j=0τ(y)1|h(fjy)h(fjy)|d¯(y,y)+γ^s(y,y),subscript𝑦subscriptsuperscript𝑦superscriptsubscript𝑗0𝜏𝑦1superscript𝑓𝑗𝑦superscript𝑓𝑗superscript𝑦much-less-than¯𝑑𝑦superscript𝑦superscript^𝛾𝑠𝑦superscript𝑦|h_{\ell}(y)-h_{\ell}(y^{\prime})|\leq\sum_{j=0}^{\tau(y)-1}|h(f^{j}y)-h(f^{j}y^{\prime})|\ll\bar{d}(y,y^{\prime})+\hat{\gamma}^{s(y,y^{\prime})}, (9.17)

for 0τ(y)0𝜏𝑦0\leq\ell\leq\tau(y).

Fix y,yYj𝑦superscript𝑦subscript𝑌𝑗y,y^{\prime}\in Y_{j} for some j1𝑗1j\geq 1, and u[0,φ(y)][0,φ(y)]𝑢0𝜑𝑦0𝜑superscript𝑦u\in[0,\varphi(y)]\cap[0,\varphi(y^{\prime})]. We will focus on choosing the appropriate t𝑡t and obtaining the relevant estimates. The choice of tsuperscript𝑡t^{\prime} is analogous. Recall the notation d¯(y,z)=d(y,z)1/4¯𝑑𝑦𝑧𝑑superscript𝑦𝑧14\bar{d}(y,z)=d(y,z)^{1/4} and note that d¯(y,z)d¯(y,y)much-less-than¯𝑑𝑦𝑧¯𝑑𝑦superscript𝑦\bar{d}(y,z)\ll\bar{d}(y,y^{\prime}).

First adjustment. As in the proof of Proposition 9.5, we arrive at Tuy=Trfysubscript𝑇𝑢𝑦subscript𝑇𝑟superscript𝑓𝑦T_{u}y=T_{r}f^{\ell}y and Tt1z=Tr1fzsubscript𝑇subscript𝑡1𝑧subscript𝑇subscript𝑟1superscript𝑓𝑧T_{t_{1}}z=T_{r_{1}}f^{\ell}z for the same 0τ(y)10𝜏𝑦10\leq\ell\leq\tau(y)-1, and such that |ut1|d¯(y,z)much-less-than𝑢subscript𝑡1¯𝑑𝑦𝑧|u-t_{1}|\ll\bar{d}(y,z) and |rr1|d¯(y,z)much-less-than𝑟subscript𝑟1¯𝑑𝑦𝑧|r-r_{1}|\ll\bar{d}(y,z). Indeed, a priori we have Tuy=Trfysubscript𝑇𝑢𝑦subscript𝑇𝑟superscript𝑓𝑦T_{u}y=T_{r}f^{\ell}y and Tuz=Trfzsubscript𝑇𝑢𝑧subscript𝑇superscript𝑟superscript𝑓superscript𝑧T_{u}z=T_{r^{\prime}}f^{\ell^{\prime}}z, where, as infh>0infimum0\inf h>0, shrinking diamYdiam𝑌\operatorname{diam}Y if needed, (9.17) implies ||1superscript1|\ell-\ell^{\prime}|\leq 1. If =superscript\ell=\ell^{\prime}, then let t1=usubscript𝑡1𝑢t_{1}=u, r1=rsubscript𝑟1superscript𝑟r_{1}=r^{\prime}, and |rr1|d¯(y,z)much-less-than𝑟subscript𝑟1¯𝑑𝑦𝑧|r-r_{1}|\ll\bar{d}(y,z) follows from (9.17). If =1superscript1\ell^{\prime}=\ell-1, then Tuz=Trfzsubscript𝑇𝑢𝑧subscript𝑇superscript𝑟superscript𝑓𝑧T_{u}z=T_{-r^{*}}f^{\ell}z, where r=h(f1z)r[0,h(f1z)]superscript𝑟superscript𝑓1𝑧superscript𝑟0superscript𝑓1𝑧r^{*}=h(f^{\ell-1}z)-r^{\prime}\in[0,h(f^{\ell-1}z)]. Note that u=h(y)+r=h(z)r𝑢subscript𝑦𝑟subscript𝑧superscript𝑟u=h_{\ell}(y)+r=h_{\ell}(z)-r^{*}, hence r+r=h(z)h(y)d¯(y,z)𝑟superscript𝑟subscript𝑧subscript𝑦much-less-than¯𝑑𝑦𝑧r+r^{*}=h_{\ell}(z)-h_{\ell}(y)\ll\bar{d}(y,z). Let t1=u+r+rsubscript𝑡1𝑢𝑟superscript𝑟t_{1}=u+r+r^{*}, so that |t1u|d¯(y,z)much-less-thansubscript𝑡1𝑢¯𝑑𝑦𝑧|t_{1}-u|\ll\bar{d}(y,z) and r1=rsubscript𝑟1𝑟r_{1}=r as Tt1z=Trfzsubscript𝑇subscript𝑡1𝑧subscript𝑇𝑟superscript𝑓𝑧T_{t_{1}}z=T_{r}f^{\ell}z. Note that we do not claim anything about d(Tuy,Tt1z)𝑑subscript𝑇𝑢𝑦subscript𝑇subscript𝑡1𝑧d(T_{u}y,T_{t_{1}}z) at this point.

Second adjustment. For brevity, introduce y^=fy^𝑦superscript𝑓𝑦\hat{y}=f^{\ell}y and z^=fz^𝑧superscript𝑓𝑧\hat{z}=f^{\ell}z. We have

Tuy=Try^=Tsf~ky^,Tt1z=Tr1z^=Tsf~kz^,formulae-sequencesubscript𝑇𝑢𝑦subscript𝑇𝑟^𝑦subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑡1𝑧subscript𝑇subscript𝑟1^𝑧subscript𝑇superscript𝑠superscript~𝑓superscript𝑘^𝑧T_{u}y=T_{r}\hat{y}=T_{s}\tilde{f}^{k}\hat{y},\qquad T_{t_{1}}z=T_{r_{1}}\hat{z}=T_{s^{\prime}}\tilde{f}^{k^{\prime}}\hat{z},

for some 0k,kr~(y^)1formulae-sequence0𝑘superscript𝑘~𝑟^𝑦10\leq k,k^{\prime}\leq\tilde{r}(\hat{y})-1 (note that r~(y^)=r~(z^)~𝑟^𝑦~𝑟^𝑧\tilde{r}(\hat{y})=\tilde{r}(\hat{z})), s[0,h~(f~ky^))𝑠0~superscript~𝑓𝑘^𝑦s\in[0,\tilde{h}(\tilde{f}^{k}\hat{y})) and s[0,h~(f~kz^))superscript𝑠0~superscript~𝑓superscript𝑘^𝑧s^{\prime}\in[0,\tilde{h}(\tilde{f}^{k^{\prime}}\hat{z})). Note that by (9.13), (9.14) and (9.3), for any 0kr~(y^)10𝑘~𝑟^𝑦10\leq k\leq\tilde{r}(\hat{y})-1, we have

d(f~ky^,f~kz^)d(y^,z^)d(y,z),hence|h~k(y^)h~k(z^)|d¯(y,z),formulae-sequencemuch-less-than𝑑superscript~𝑓𝑘^𝑦superscript~𝑓𝑘^𝑧𝑑^𝑦^𝑧much-less-than𝑑𝑦𝑧much-less-thanhencesubscript~𝑘^𝑦subscript~𝑘^𝑧¯𝑑𝑦𝑧d(\tilde{f}^{k}\hat{y},\tilde{f}^{k}\hat{z})\ll d(\hat{y},\hat{z})\ll d(y,z),\quad\text{hence}\quad|\tilde{h}_{k}(\hat{y})-\tilde{h}_{k}(\hat{z})|\ll\bar{d}(y,z), (9.18)

where we have used (9.11). We distinguish three cases: k=k𝑘superscript𝑘k=k^{\prime}, k>k𝑘superscript𝑘k>k^{\prime} and k<k𝑘superscript𝑘k<k^{\prime}.

If k=k𝑘superscript𝑘k=k^{\prime}, (9.18) along with |rr1|d¯(y,z)much-less-than𝑟subscript𝑟1¯𝑑𝑦𝑧|r-r_{1}|\ll\bar{d}(y,z) implies |ss|d¯(y,z)much-less-than𝑠superscript𝑠¯𝑑𝑦𝑧|s-s^{\prime}|\ll\bar{d}(y,z). But then, again by (9.18), (9.13) and (9.14), we have

d(Tuy,Tt1z)=d(Tsf~ky^,Tsf~kz^)d¯(y,z).𝑑subscript𝑇𝑢𝑦subscript𝑇subscript𝑡1𝑧𝑑subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇superscript𝑠superscript~𝑓𝑘^𝑧much-less-than¯𝑑𝑦𝑧d(T_{u}y,T_{t_{1}}z)=d(T_{s}\tilde{f}^{k}\hat{y},T_{s^{\prime}}\tilde{f}^{k}\hat{z})\ll\bar{d}(y,z).

As |ut1|d¯(y,z)much-less-than𝑢subscript𝑡1¯𝑑𝑦𝑧|u-t_{1}|\ll\bar{d}(y,z), we can fix t=t1𝑡subscript𝑡1t=t_{1}.

If k>k𝑘superscript𝑘k>k^{\prime}, we prefer to represent our points as

Tuy=Try^=Tsf~ky^,Tt1z=Tr1z^=Ts1f~kz^formulae-sequencesubscript𝑇𝑢𝑦subscript𝑇𝑟^𝑦subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑡1𝑧subscript𝑇subscript𝑟1^𝑧subscript𝑇subscript𝑠1superscript~𝑓𝑘^𝑧T_{u}y=T_{r}\hat{y}=T_{s}\tilde{f}^{k}\hat{y},\qquad T_{t_{1}}z=T_{r_{1}}\hat{z}=T_{-s_{1}}\tilde{f}^{k}\hat{z}

for some s1>0subscript𝑠10s_{1}>0. Now by (9.18) and as |rr1|d¯(y,z)much-less-than𝑟subscript𝑟1¯𝑑𝑦𝑧|r-r_{1}|\ll\bar{d}(y,z), we have s+s1d¯(y,z)much-less-than𝑠subscript𝑠1¯𝑑𝑦𝑧s+s_{1}\ll\bar{d}(y,z). Define

s2=min(s,h~(f~kz^)/2,h~(f~ky^)/2),r2=s2+s1+r1,t=s2+s1+t1.formulae-sequencesubscript𝑠2𝑠~superscript~𝑓𝑘^𝑧2~superscript~𝑓𝑘^𝑦2formulae-sequencesubscript𝑟2subscript𝑠2subscript𝑠1subscript𝑟1𝑡subscript𝑠2subscript𝑠1subscript𝑡1s_{2}=\min(s,\tilde{h}(\tilde{f}^{k}\hat{z})/2,\tilde{h}(\tilde{f}^{k}\hat{y})/2),\quad r_{2}=s_{2}+s_{1}+r_{1},\quad t=s_{2}+s_{1}+t_{1}.

Then Ttz=Ts2f~kz^subscript𝑇𝑡𝑧subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑧T_{t}z=T_{s_{2}}\tilde{f}^{k}\hat{z}, where s2[0,h~(f~ky^))[0,h~(f~kz^))subscript𝑠20~superscript~𝑓𝑘^𝑦0~superscript~𝑓𝑘^𝑧s_{2}\in[0,\tilde{h}(\tilde{f}^{k}\hat{y}))\cap[0,\tilde{h}(\tilde{f}^{k}\hat{z})) and

|ss2|ss+s1d¯(y,z).𝑠subscript𝑠2𝑠𝑠subscript𝑠1much-less-than¯𝑑𝑦𝑧|s-s_{2}|\leq s\leq s+s_{1}\ll\bar{d}(y,z).

Hence

d(Tuy,Ttz)=d(Tsf~ky^,Ts2f~kz^)d(Ts2f~ky^,Ts2f~kz^)+d(Tsf~ky^,Ts2f~ky^),𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧𝑑subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑧𝑑subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑧𝑑subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑦d(T_{u}y,T_{t}z)=d(T_{s}\tilde{f}^{k}\hat{y},T_{s_{2}}\tilde{f}^{k}\hat{z})\leq d(T_{s_{2}}\tilde{f}^{k}\hat{y},T_{s_{2}}\tilde{f}^{k}\hat{z})+d(T_{s}\tilde{f}^{k}\hat{y},T_{s_{2}}\tilde{f}^{k}\hat{y}),

where d(Tsf~ky^,Ts2f~ky^)d¯(y,z)much-less-than𝑑subscript𝑇𝑠superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑦¯𝑑𝑦𝑧d(T_{s}\tilde{f}^{k}\hat{y},T_{s_{2}}\tilde{f}^{k}\hat{y})\ll\bar{d}(y,z) by (9.12), while d(Ts2f~ky^,Ts2f~kz^)d¯(y,z)𝑑subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑦subscript𝑇subscript𝑠2superscript~𝑓𝑘^𝑧¯𝑑𝑦𝑧d(T_{s_{2}}\tilde{f}^{k}\hat{y},T_{s_{2}}\tilde{f}^{k}\hat{z})\leq\bar{d}(y,z) by (9.13), (9.14) and (9.18). Hence d(Tuy,Ttz)d¯(y,z)much-less-than𝑑subscript𝑇𝑢𝑦subscript𝑇𝑡𝑧¯𝑑𝑦𝑧d(T_{u}y,T_{t}z)\ll\bar{d}(y,z), as desired. On the other hand |tt1|=s1+s2s1+sd¯(y,z)𝑡subscript𝑡1subscript𝑠1subscript𝑠2subscript𝑠1𝑠much-less-than¯𝑑𝑦𝑧|t-t_{1}|=s_{1}+s_{2}\leq s_{1}+s\ll\bar{d}(y,z), and as we have already controlled |t1u|subscript𝑡1𝑢|t_{1}-u|, we have |tu|d¯(y,z)much-less-than𝑡𝑢¯𝑑𝑦𝑧|t-u|\ll\bar{d}(y,z).

The case when k<k𝑘superscript𝑘k<k^{\prime} can be treated analogously. The choice of tsuperscript𝑡t^{\prime} goes along similar lines, so we omit the details. ∎

Theorem 9.13

Consider a semi-dispersing Lorentz flow or the billiard flow in a Bunimovich stadium. Let η(0,1]𝜂01\eta\in(0,1]. There exists m1𝑚1m\geq 1 such that ρv,w(t)=O(t1)subscript𝜌𝑣𝑤𝑡𝑂superscript𝑡1\rho_{v,w}(t)=O(t^{-1}) for all vCη(M)C0,η(M)𝑣superscript𝐶𝜂𝑀superscript𝐶0𝜂𝑀v\in C^{\eta}(M)\cap C^{0,\eta}(M) and wCη,m(M)𝑤superscript𝐶𝜂𝑚𝑀w\in C^{\eta,m}(M) (and more generally for the class of observables defined in Corollary 9.6).

Proof.

It follows from Lemma 8.3 and Corollary 9.11 that condition (H) is satisfied. Hence by Corollary 8.1(a), the suspension flow Ft:YφYφ:subscript𝐹𝑡superscript𝑌𝜑superscript𝑌𝜑F_{t}:Y^{\varphi}\to Y^{\varphi} is a Gibbs-Markov flow as defined in Section 6. The conclusions of Corollary 9.6 follow from Propositions 9.10 and 9.12. Hence the flows and observables are dynamically Hölder (Definition 7.6).

For the tail estimate on φ𝜑\varphi, introduce τ~:Y+:~𝜏𝑌superscript\tilde{\tau}:Y\to{\mathbb{Z}}^{+}, τ~(y)=min{n1:f~nyY}~𝜏𝑦:𝑛1superscript~𝑓𝑛𝑦𝑌\tilde{\tau}(y)=\min\{n\geq 1:\tilde{f}^{n}y\in Y\}. Note that suph~<supremum~\sup\tilde{h}<\infty, and φ(y)=k=0τ~(y)1h~(f~ky)τ~(y)suph~𝜑𝑦superscriptsubscript𝑘0~𝜏𝑦1~superscript~𝑓𝑘𝑦~𝜏𝑦supremum~\varphi(y)=\sum_{k=0}^{\tilde{\tau}(y)-1}\tilde{h}(\tilde{f}^{k}y)\leq\tilde{\tau}(y)\sup\tilde{h} . Also it is shown in [15] (both for the semi-dispersing examples and for stadia) that μ(τ~>n)=O(n2)𝜇~𝜏𝑛𝑂superscript𝑛2\mu(\tilde{\tau}>n)=O(n^{-2}). Hence μ(φ>t)μ(τ~suph~>t)=O(t2)𝜇𝜑𝑡𝜇~𝜏supremum~𝑡𝑂superscript𝑡2\mu(\varphi>t)\leq\mu(\tilde{\tau}\sup\tilde{h}>t)=O(t^{-2}).

Finally, to exclude approximate eigenfunctions, we may appeal as at the end of Section 9.2 to the contact structure which the billiard examples have in common. The result now follows from Corollary 8.1(b). ∎

9.4 Lower bounds

In this subsection, we show that it is impossible to improve on the error rate O(t1)𝑂superscript𝑡1O(t^{-1}) for infinite horizon Lorentz gases, semidispersing Lorentz flows, and Bunimovich stadia. The following result is based on [5, Corollary 1.3].

Proposition 9.14

Let vL2(M)𝑣superscript𝐿2𝑀v\in L^{2}(M) with Mv𝑑μM=0subscript𝑀𝑣differential-dsubscript𝜇𝑀0\int_{M}v\,d\mu_{M}=0. Suppose that ρv,v(t)=o(t1)subscript𝜌𝑣𝑣𝑡𝑜superscript𝑡1\rho_{v,v}(t)=o(t^{-1}). Then |0tvTs𝑑s|2=o((tlogt)1/2)subscriptsuperscriptsubscript0𝑡𝑣subscript𝑇𝑠differential-d𝑠2𝑜superscript𝑡𝑡12|\int_{0}^{t}v\circ T_{s}\,ds|_{2}=o((t\log t)^{1/2}).

Proof.

Let vt=0tvTs𝑑ssubscript𝑣𝑡superscriptsubscript0𝑡𝑣subscript𝑇𝑠differential-d𝑠v_{t}=\int_{0}^{t}v\circ T_{s}\,ds. Then

Mvt2𝑑μMsubscript𝑀superscriptsubscript𝑣𝑡2differential-dsubscript𝜇𝑀\displaystyle\int_{M}v_{t}^{2}\,d\mu_{M} =0t0tMvTrvTs𝑑μM𝑑r𝑑s=20t0sMvvTsr𝑑μM𝑑r𝑑sabsentsuperscriptsubscript0𝑡superscriptsubscript0𝑡subscript𝑀𝑣subscript𝑇𝑟𝑣subscript𝑇𝑠differential-dsubscript𝜇𝑀differential-d𝑟differential-d𝑠2superscriptsubscript0𝑡superscriptsubscript0𝑠subscript𝑀𝑣𝑣subscript𝑇𝑠𝑟differential-dsubscript𝜇𝑀differential-d𝑟differential-d𝑠\displaystyle=\int_{0}^{t}\int_{0}^{t}\int_{M}v\circ T_{r}\,v\circ T_{s}\,d\mu_{M}\,dr\,ds=2\int_{0}^{t}\int_{0}^{s}\int_{M}v\,v\circ T_{s-r}\,d\mu_{M}\,dr\,ds
=20t0sρv,v(r)𝑑r𝑑s=20trtρv,v(r)𝑑s𝑑r2t0tρv,v(r)𝑑r.absent2superscriptsubscript0𝑡superscriptsubscript0𝑠subscript𝜌𝑣𝑣𝑟differential-d𝑟differential-d𝑠2superscriptsubscript0𝑡superscriptsubscript𝑟𝑡subscript𝜌𝑣𝑣𝑟differential-d𝑠differential-d𝑟2𝑡superscriptsubscript0𝑡subscript𝜌𝑣𝑣𝑟differential-d𝑟\displaystyle=2\int_{0}^{t}\int_{0}^{s}\rho_{v,v}(r)\,dr\,ds=2\int_{0}^{t}\int_{r}^{t}\rho_{v,v}(r)\,ds\,dr\leq 2t\int_{0}^{t}\rho_{v,v}(r)\,dr.

By the assumption on ρv,vsubscript𝜌𝑣𝑣\rho_{v,v}, we obtain |vt|22=o(tlogt)superscriptsubscriptsubscript𝑣𝑡22𝑜𝑡𝑡|v_{t}|_{2}^{2}=o(t\log t). ∎

In the case of the planar infinite horizon Lorentz gas, Szász & Varjú [29] showed that (tlogt)1/20tvTs𝑑ssuperscript𝑡𝑡12superscriptsubscript0𝑡𝑣subscript𝑇𝑠differential-d𝑠(t\log t)^{-1/2}\int_{0}^{t}v\circ T_{s}\,ds converges in distribution to a nondegenerate normal distribution for typical Hölder mean zero observables v𝑣v. The result applies equally to semidispersing Lorentz flows. Similarly, in the case of Bunimovich stadia by Bálint & Gouëzel [5, Corollary 1.6]. In particular, (tlogt)1/2|0tvTs𝑑s|2↛0↛superscript𝑡𝑡12subscriptsuperscriptsubscript0𝑡𝑣subscript𝑇𝑠differential-d𝑠20(t\log t)^{-1/2}|\int_{0}^{t}v\circ T_{s}\,ds|_{2}\not\to 0. Hence by Proposition 9.14, an upper bound of the type o(t1)𝑜superscript𝑡1o(t^{-1}) is impossible and so the upper bound in Theorems 9.1 and 9.13 is optimal.

Remark 9.15

There is also the possibility of obtaining an asymptotic expression of the form

ρv,w(t)=ct1+O(t(2ϵ)),subscript𝜌𝑣𝑤𝑡𝑐superscript𝑡1𝑂superscript𝑡2italic-ϵ\rho_{v,w}(t)=ct^{-1}+O(t^{-(2-\epsilon)}), (9.19)

(ϵ>0italic-ϵ0\epsilon>0 arbitrarily small, c>0𝑐0c>0) for certain classes of observables v,w𝑣𝑤v,w. Such results are obtained in [27] in cases where there is a first return to a uniformly hyperbolic map f:XX:𝑓𝑋𝑋f:X\to X. The first return map in the examples considered here is nonuniformly hyperbolic, modelled by a Young tower with exponential tails, so [27] does not apply directly. In a recent preprint, [10] have announced the existence of a uniformly hyperbolic first return. This combined with [27] may yield the asymptotic (9.19). (Interestingly, the class of observables in (9.19) would be disjoint from the class of observables covered by Proposition 9.14.)

Appendix A Condition (7.4)

In this appendix, we verify that condition (7.4) holds in the abstract framework of [30]. For this purpose, we switch to the notation of [30].

Proposition A.1

Let f:ΛΛ:𝑓ΛΛf:\Lambda\to\Lambda be an injective transformation satisfying the abstract set up in [30, Section 1]: specifically (P1), the second part of (P2), property (iii) of the separation time s0subscript𝑠0s_{0}, and (P4)(a).

Let xΛi𝑥subscriptΛ𝑖x\in\Lambda_{i}, i1𝑖1i\geq 1. Then fRi(γu(x)Λi)=γu(fRix)Λsuperscript𝑓subscript𝑅𝑖superscript𝛾𝑢𝑥subscriptΛ𝑖superscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥Λf^{R_{i}}(\gamma^{u}(x)\cap\Lambda_{i})=\gamma^{u}(f^{R_{i}}x)\cap\Lambda.

Proof.

It follows from injectivity of f𝑓f and hence fRisuperscript𝑓subscript𝑅𝑖f^{R_{i}}, as well as (P2), that

fRi(γu(x)Λi)=fRiγu(x)fRiΛiγu(fRix)fRiΛi.superscript𝑓subscript𝑅𝑖superscript𝛾𝑢𝑥subscriptΛ𝑖superscript𝑓subscript𝑅𝑖superscript𝛾𝑢𝑥superscript𝑓subscript𝑅𝑖subscriptΛ𝑖superset-ofsuperscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥superscript𝑓subscript𝑅𝑖subscriptΛ𝑖f^{R_{i}}(\gamma^{u}(x)\cap\Lambda_{i})=f^{R_{i}}\gamma^{u}(x)\cap f^{R_{i}}\Lambda_{i}\supset\gamma^{u}(f^{R_{i}}x)\cap f^{R_{i}}\Lambda_{i}. (A.1)

Recall from (P1) that we have the local product structure Λ=(kKuγku)(Ksγs)Λsubscript𝑘superscript𝐾𝑢superscriptsubscript𝛾𝑘𝑢subscriptsuperscript𝐾𝑠superscriptsubscript𝛾𝑠\Lambda=\big{(}\bigcup_{k\in K^{u}}\gamma_{k}^{u}\big{)}\cap\big{(}\bigcup_{\ell\in K^{s}}\gamma_{\ell}^{s}\big{)}. By (P2), fRiΛisuperscript𝑓subscript𝑅𝑖subscriptΛ𝑖f^{R_{i}}\Lambda_{i} is a u𝑢u-subset of ΛΛ\Lambda which means that fRiΛi=(kKiuγku)(Ksγs)superscript𝑓subscript𝑅𝑖subscriptΛ𝑖subscript𝑘superscriptsubscript𝐾𝑖𝑢superscriptsubscript𝛾𝑘𝑢subscriptsuperscript𝐾𝑠superscriptsubscript𝛾𝑠f^{R_{i}}\Lambda_{i}=\big{(}\bigcup_{k\in K_{i}^{u}}\gamma_{k}^{u}\big{)}\cap\big{(}\bigcup_{\ell\in K^{s}}\gamma_{\ell}^{s}\big{)} for some subset KiuKusuperscriptsubscript𝐾𝑖𝑢superscript𝐾𝑢K_{i}^{u}\subset K^{u}. Hence γkuΛ=γku(Ksγs)=γkufRiΛisuperscriptsubscript𝛾𝑘𝑢Λsuperscriptsubscript𝛾𝑘𝑢subscriptsuperscript𝐾𝑠superscriptsubscript𝛾𝑠superscriptsubscript𝛾𝑘𝑢superscript𝑓subscript𝑅𝑖subscriptΛ𝑖\gamma_{k}^{u}\cap\Lambda=\gamma_{k}^{u}\cap\big{(}\bigcup_{\ell\in K^{s}}\gamma_{\ell}^{s}\big{)}=\gamma_{k}^{u}\cap f^{R_{i}}\Lambda_{i} for all kKiu𝑘superscriptsubscript𝐾𝑖𝑢k\in K_{i}^{u}. Also, γkufRiΛi=superscriptsubscript𝛾𝑘𝑢superscript𝑓subscript𝑅𝑖subscriptΛ𝑖\gamma_{k}^{u}\cap f^{R_{i}}\Lambda_{i}=\emptyset for all kKiu𝑘superscriptsubscript𝐾𝑖𝑢k\not\in K_{i}^{u}.

Now, γu(fRix)fRiΛisuperscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥superscript𝑓subscript𝑅𝑖subscriptΛ𝑖\gamma^{u}(f^{R_{i}}x)\cap f^{R_{i}}\Lambda_{i}\neq\emptyset (it contains fRixsuperscript𝑓subscript𝑅𝑖𝑥f^{R_{i}}x) so it follows from the above considerations that γu(fRix)Λ=γu(fRix)fRiΛisuperscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥Λsuperscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥superscript𝑓subscript𝑅𝑖subscriptΛ𝑖\gamma^{u}(f^{R_{i}}x)\cap\Lambda=\gamma^{u}(f^{R_{i}}x)\cap f^{R_{i}}\Lambda_{i}. Combining this with (A.1),

fRi(γu(x)Λi)γu(fRix)Λ.superscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥Λsuperscript𝑓subscript𝑅𝑖superscript𝛾𝑢𝑥subscriptΛ𝑖f^{R_{i}}(\gamma^{u}(x)\cap\Lambda_{i})\supset\gamma^{u}(f^{R_{i}}x)\cap\Lambda. (A.2)

It remains to prove the reverse inclusion, so suppose that yγu(x)Λi𝑦superscript𝛾𝑢𝑥subscriptΛ𝑖y\in\gamma^{u}(x)\cap\Lambda_{i}. By (P1), there exists zγu(fRix)γs(fRiy)Λsuperscript𝑧superscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥superscript𝛾𝑠superscript𝑓subscript𝑅𝑖𝑦Λz^{*}\in\gamma^{u}(f^{R_{i}}x)\cap\gamma^{s}(f^{R_{i}}y)\subset\Lambda. By (A.2), z=fRizsuperscript𝑧superscript𝑓subscript𝑅𝑖𝑧z^{*}=f^{R_{i}}z for some zγu(x)Λi𝑧superscript𝛾𝑢𝑥subscriptΛ𝑖z\in\gamma^{u}(x)\cap\Lambda_{i}.

Since zsuperscript𝑧z^{*} and fRiysuperscript𝑓subscript𝑅𝑖𝑦f^{R_{i}}y lie in the same stable disk it follows from property (iii) of the separation time that s0(z,fRiy)=subscript𝑠0superscript𝑧superscript𝑓subscript𝑅𝑖𝑦s_{0}(z^{*},f^{R_{i}}y)=\infty. Using property (iii) once more, s0(z,y)s0(z,fRiy)=subscript𝑠0𝑧𝑦subscript𝑠0superscript𝑧superscript𝑓subscript𝑅𝑖𝑦s_{0}(z,y)\geq s_{0}(z^{*},f^{R_{i}}y)=\infty. But zγu(x)=γu(y)𝑧subscript𝛾𝑢𝑥subscript𝛾𝑢𝑦z\in\gamma_{u}(x)=\gamma_{u}(y) so (P4)(a) implies that d(z,y)Cαs0(z,y)=0𝑑𝑧𝑦𝐶superscript𝛼subscript𝑠0𝑧𝑦0d(z,y)\leq C\alpha^{s_{0}(z,y)}=0. Hence fRiy=fRiz=zγu(fRix)superscript𝑓subscript𝑅𝑖𝑦superscript𝑓subscript𝑅𝑖𝑧superscript𝑧superscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥f^{R_{i}}y=f^{R_{i}}z=z^{*}\in\gamma^{u}(f^{R_{i}}x). This shows that fRi(γu(x)Λi)γu(fRix)Λsuperscript𝑓subscript𝑅𝑖superscript𝛾𝑢𝑥subscriptΛ𝑖superscript𝛾𝑢superscript𝑓subscript𝑅𝑖𝑥Λf^{R_{i}}(\gamma^{u}(x)\cap\Lambda_{i})\subset\gamma^{u}(f^{R_{i}}x)\cap\Lambda completing the proof. ∎

Acknowledgements

The research of PB was supported in part by Hungarian National Foundation for Scientific Research (NKFIH OTKA) grants K104745 and K123782. OB was supported in part by EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS). The research of IM was supported in part by a European Advanced Grant StochExtHomog (ERC AdG 320977).

We are grateful to the referees for very helpful comments which led to many clarifications and corrections.

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